The Torah tells us that Moshe informed the people that one of the materials that was needed for the construction and preparation of the Mishkan was something called "Techeiles" (תכלת), which they were asked to donate (שמות ל"ה:ו'). It is clear from the commentary of Rashi earlier in the Torah (שם כ"ה:ד' בד"ה ותכלת) that the reference is to wool which was dyed a certain color, as indicated as well by the Gemara in Yevamos (דף ד:, ועיין שם ברש"י ד"ה ותכלת ובתוד"ה מדשש); Techeiles is the name of that particular color. The Rambam (פרק ב' מהל' ציצית הלכה א' ובפרק ח' מהל' כלי המקדש הלכה י"ג) writes that whenever the Torah speaks of Techeiles, it is referring to wool that is dyed this particular color. According to the Pesukim in the Torah, including several in our Parsha, Techeiles was used for the tapestries that covered the Mishkan (שמות כ"ו:א', ל"ו:ח'), for the loops at the edge of some of the tapestries (שם כ"ו:ד', ל"ו:י"א), for the Paroches, which was the partition between different areas within the Mishkan (שם כ"ו:ל"א, ל"ו:ל"ה), and for the drape at the gate of the Mishkan's courtyard (שם כ"ז:ט"ז, ל"ח:י"ח); Techeiles was also used for several of the Bigdei Kehunah, including the Eiphod, a kind of apron (שם כ"ח:ו', ל"ט:ב'), the Choshen, that is, the breastplate (שם כ"ח:ט"ו, ל"ט:ח'), the Me'il, a kind of robe שם כ"ח:ל"א, ל"ט:כ"ב() and the pomegranate-shaped spheres that hung on its bottom (שם כ"ח:ל"ג, ל"ט:כ"ד), the Tzitz, that is, the headplate, which had a thread made of this material fastened to it (שם כ"ח:ל"ז, ל"ט:ל"א), and the Avneit, a kind of belt (שם ל"ט:כ"ט) worn by the Kohein Gadol, and, perhaps as well, the Avneit worn by the ordinary Kohanim (עיין בגמ' יומא דף ו. וברש"י שם בד"ה אמר ובתוד"ה אמר שם, ובדף י"ב: שם, וברמב"ם בפרק ח' מהל' כלי המקדש הלכה א' ובמשנה למלך להלכה ב' שם). The Pesukim later in the Torah indicate that garments made of Techeiles were also used to cover various vessels of the Mishkan, including the Aron, the Shulchan, the Menorah, the inner Mizbeiach, and certain other utensils, while they were being transported (במדבר ד':ו'-י"ב). It must be stressed, however, that in addition to its usage in the context of the construction of and the service in the Mishkan, Techeiles is also mentioned in connection with the Mitzvah of Tzitzis, concerning which the Torah states (שם ט"ו:ל"ח) that one must include a Techeiles colored thread with the Tzitzis found on each corner of one's garment.
The Mishnah in Menachos (דף ל"ח.) states regarding Tzitzis that the absence of this Techeiles thread does not invalidate the Tzitzis; the Gemara which follows, however (ובעמוד ב' שם שם), notes that this may not be accepted by all authorities. The Baal HaMaor in Shabbos ( ומ"ש הרי"ףדף י"ב. בדפי הרי"ף בד"ה), for example, thus rules that Tzitzis are not valid without the Techeiles thread, a view which he says is also accepted by the Behag, and which is presented as well in the Teshuvos of the Raavad (שו"ת תמים דעים סימן שכ"ד); Rav Yerucham Perlow, in his commentary on the Sefer HaMitzvos of Rabbeinu Saadyah Gaon (מצות עשה ז', דף צ"ה. בד"ה ומעתה), suggests that that Gaon may also have held that way. The more widely accepted view, however, is that the Techeiles thread is not absolutely necessary, as indicated by Rashi in Menachos there (שם בד"ה התכלת), as well as by the Rosh, both in the Hilchos Tzitzis section of his Halachos Ketanos, printed in the back of Maseches Menachos (סימן ז'), and in a Teshuvah (שו"ת הרא"ש כלל ב' סימן ט'), the Rambam, both in his Sefer HaMitzvos (מצות עשה י"ד) and in his Mishneh Torah (פרק א' מהל' ציצית הלכה ד'), the Ramban in Shabbos (מלחמת ה' בדף י"ב. בדפי הרי"ף בא"ד שם), the Rashba (הרשב"א חלק ד' סימן רנ"ט ובחלק ז' סימן שמ"ה שו"ת) and others. The Pri Megadim (פתיחה להל' ציצית) discusses this issue and some of its ramifications at length; it is clear, however, that it is certainly preferable to have the Techeiles thread on one's Tzitzis, if Techeiles is indeed available, as stated clearly by the Midrash in BeMidbar Rabbah (פרשה י"ז סימן ז') and by the Midrash Tanchuma in Parshas Shelach (אות ט"ו).
The question is, however, exactly what color this Techeiles is, and how it is produced. The Gemara in Menachos (דף מ"ג:, ועיין שם בתוד"ה ה"ג רקיע), as well as the aforementioned Midrash Tanchuma in Parshas Shelach (שם), among several other sources, states that the color of Techeiles is similar to the color of the sea, which in turn is similar to the color of the sky. This idea also appears in the Gemara in Sotah (דף י"ז.); Rashi there (שם בד"ה שהתכלת) stresses that the comparison to the sky is an indirect one, and that Techeiles is more like the color of the sea, a view perhaps prompted by the connection made by the Gemara (שם) to the color of sapphire, a connection also presented by the Gemara in Chulin (דף פ"ט., ועיין שם ברש"י בד"ה שהתכלת), among other places. The Klei Yakar, in his commentary on the Torah (לבמדבר שם בד"ה ועשו), stresses that these comparisons are very general. The Midrash in BeMidbar Rabbah (פרשה ד' סימן י"ג) and elsewhere, however, indicates that the comparison to the sky is in fact a direct one; the Rambam (פרק ב' מהל' ציצית הלכה א' ובהל' כלי המקדש שם הלכה י"ג) writes that Techeiles is indeed similar to the color of the sky, noting, though, that the reference is to a clear noon-time sky, a view apparently accepted by Rabbeinu Bechaya as well (לשמות כ"ח:ו' בד"ה ועשו), and Techeiles would thus be most precisely defined as "sky-blue." Rashi, however, commenting in the name of Rabbi Moshe HaDarshan on a Posuk later in the Torah (במדבר שם פסוק מ"א בד"ה פתיל), indicates that Techeiles is the color of the sky as it starts to get dark towards evening, and Rabbeinu Saadyah Gaon, in his commentary on a Posuk earlier in the Torah (שמות כ"ה:ד' בד"ה ותכלת, ועיין שם בהערת הרב יוסף קאפח מספר 2), apparently agrees; the Tosafos Yom Tov, commenting on a Mishnah in Kilayim (פרק ב' משנה ה' בד"ה ספיחי), writes that Techeiles is a dark color, bringing several proofs.
Rashi himself, though, commenting on that same Posuk earlier in the Torah (שמות שם בד"ה ותכלת), states that Techeiles has a greenish color, a position he echoes when commenting on a Posuk later in the Torah (במדבר שם פסוק ל"ח בד"ה תכלת), and in his commentary on a Mishnah in Berachos (דף ט: בד"ה תכלת) and on a Gemara there (דף נ"ז: בד"ה תכלת). This view is also accepted by Ibn Ezra (לשמות שם בד"ה תכלת), and it is perhaps based on the idea found in, among other places, the Midrash in BeMidbar Rabbah (פרשה י"ד סימן י') and in the Yerushalmi in Berachos (פרק א' הלכה ב', דף ז:) which states that Techeiles is similar to grass; this approach is also discussed in Tosafos in Sukkah (הירוק דף ל"א: בד"ה), where the color blue is actually mentioned in a footnote on the side. The Radak, in his commentary on the Torah (בד"ה תכלת לבמדבר שם), mentions the color "ultramarine," a deep-blue color, while the Chizkuni (לשמות כ"ד:י' בד"ה כמעשה) specifically mentions blue to explain the color of Techeiles, as does the Tiferes Yisrael, in one of his introductory essays on the Mishnayos in Seder Moed entitled "קופת הרוכלים: כללי בגדי קדש של כהונה" (בד"ה בענין). It must be noted that the Gemara in Menachos (דף מ"א:) refers to a dye derived from a plant called "קלא אילן," which Rashi (שם בד"ה חוץ) explains is very similar to Techeiles, and the Nimukei Yosef in Bava Metzia (דף ל"ד. בדפי הרי"ף בד"ה אילן) identifies this as indigo, adding that this is also the opinion of the Aruch (ערך קלא אילן). There seems to be little doubt that Techeiles is indeed a shade of blue; as for the aforementioned sources which indicate that it has a greenish appearance, it is noteworthy that Dr. Yisrael Ziderman, a contemporary scholar (מאמר "לחידוש מצות תכלת בציצית", נדפס בספר "תחומין" תשמ"ח), points out that during the processing of the dye to yield a blue or purple color, the dye changes colors, turning green for a time (when exposed to sunlight), a fact which may reconcile some of the above opinions.
The Tosefta in Menachos (פרק ט' הלכה ו') states that Techeiles may be obtained only from something known as the Chilazon, which is a living creature, as seen clearly from the Gemara in Shabbos (דף ע"ה.) which speaks of taking its life away, and the Gemara in Menachos (דף מ"ב:) describes how the dye is to be processed, after the דם, that is, a mucus-like substance, has been removed from the Chilazon. This description is elaborated upon by the Rambam (הל' ציצית שם הלכה ב'), who then adds (שם הלכה ג') that the dyeing must be done לשמה, that is, for the sake of the Mitzvah. It must be noted, though, that some hold that Techeiles need not come specifically from the Chilazon, as discussed by the Tiferes Yisrael in his aforementioned essay called "קופת הרוכלים" (בד"ה אמנם הדבר ובד"ה אמנם ובד"ה הכלל), among others; Rav Yechiel Michel Tukatchinsky (ספר עיר הקדש והמקדש, חלק ה' פרק ה' סעיף ג') discusses this matter at length. The Gemara later in Menachos (דף מ"ד.) states that the appearance of the Chilazon is similar to that of a fish; Rashi in Shabbos (דף ע"ד: בד"ה צדי) indeed identifies it as being like a fish, while Tosafos there (שם דף ע"ג: בד"ה מפרק) states that it is in fact a fish, as do the Rambam (שם הלכה ב'), the Sefer HaChinuch (מצוה שפ"ו), and others. It should be noted, though, that the Vilna Gaon, commenting on a Mishnah in Keilim (אליהו רבא לפרק י' משנה א' שם בד"ה ה"ג עצמות), explains that anything which is in the sea is identified as a kind of fish, and it is clear from Rashi in Bava Metzia (דף ס"א: בד"ה קלא אילן) and in Sanhedrin (דף צ"א. בד"ה חלזון) that the Chilazon is indeed generally found in the sea, although Rashi in Menachos (שם בד"ה ועולה) implies otherwise; this apparent contradiction is resolved in a Teshuvah presented by the Noda BeYehudah (שו"ת נודע ביהודה מהדורא תנינא חלק אורח חיים סימן ג') from his son.
According to the simple reading of the Yerushalmi in Shabbos (פרק א' הלכה ג', דף ח., ועיין בקרבן העדה שם בד"ה ופריך), the Chilazon is a boneless invertebrate, while the Midrash in Shir HaShirim Rabbah (פרשה ד' סימן כ"ג) indicates that it has a shell which grows with it; the Ran in Shabbos (חידושי הר"ן לדף ק"ז. שם בד"ה הצדן והחובל) states that its body is opaque and that it has no limbs, while the aforementioned Rashi in Sanhedrin (שם) describes it as a worm. The Gemara in Menachos (שם) also notes that the Chilazon is a very rare and therefore very expensive creature, and that its very body is somehow similar to the sea; the Gemara in Megillah (ו. דף) states that these creatures may be found in the territory of Sheivet Zevulun, along the northern coast of Eretz Yisrael. Despite all of the above signs of identification, though, it is currently difficult to definitively identify what creature the Chilazon really is, because, as already noted in the Sifrei in Devarim (שנ"ד, פרשת ברכה פיסקא י"ג פיסקא), the Chilazon even then was not easily found, and as stated by the Midrash in BeMidbar Rabbah (י"ז סימן ז' פרשה) and by the Midrash Tanchuma in Parshas Shelach (שם), Techeiles in that era was no longer available, and it is therefore not used when making Tzitzis.
In recent generations, however, efforts have been made to identify the Chilazon precisely, and, subsequently, to once again produce Techeiles. In the late 1880's, Rabbi Gershon Henoch Leiner, the Rebbe of Radzyn, embarked on a mission to find the true identity of the "lost" Chilazon, and to revive the use of Techeiles, as he describes in the first of three Seforim which he wrote in connection with this matter (מאמר שפוני טמוני חול, עמוד כ"ו). His efforts led him to conclude that the "cuttlefish," a kind of squid, is in fact the Chilazon, as explained by Rabbi Menachem Burstin, whose Sefer entitled "התכלת" includes an outline of the activities and accomplishments of the Radzyner Rebbe in this regard, and is a thorough survey of this entire subject (פרק ח' הלכה א', עמוד 152 שם). Shortly thereafter, thousands of Radzyner Chassidim were wearing Tzitzis with Techeiles, and Rabbi Leiner published two other Seforim (מאמר פתיל תכלת and מאמר עין התכלת) to further present and defend his position. In 1913, however, Rav Yitzchak Isaac HaLeivi Herzog (who later became the second Chief Rabbi of Eretz Yisrael) wrote his doctoral dissertation (in English) on the subject of "Hebrew Porphyrology," meaning, the study of purple, which, in his opinion, includes Techeiles; the major points of this work are presented in Hebrew in Rabbi Burstin's aforementioned Sefer (מאמר התכלת בישראל, נדפס כנספח לספר "התכלת"). Rav Herzog (מאמר התכלת בישראל פרק ט' סעיף ב') raises several questions against the conclusion of the Radzyner Rebbe about the identity of the Chilazon from the perspective of Halacha, and then notes (שם סעיף ג') that chemical tests revealed that the dye used by the Radzyner Chassidim was not in fact organic, but was mixed with an inorganic material yielding a color known as "Prussian blue," presumably the result of the Radzyner Rebbe having been misled by a dishonest chemist. Rav Yechiel Michel Tukatchinsky (הקדש והמקדש שם סעיף ה' אות ה' ספר עיר) also rejected the conclusions of the Radzyner Rebbe for reasons he details.
Rav Herzog himself (שם פרק י"ב) concludes that the Chilazon is in fact a species of snail called the "Janthina," and he analyzes the problems with that conclusion and proposes solutions; Rav Tukatchinsky (שם סעיף ו' אות ו') appears to accept this position. More recently, however, several researchers have expressed the preference for identifying the Chilazon as a snail called the "Murex Trunculus," a position Rav Herzog was aware of but rejected (שם פרק י"א); this view is detailed in Rabbi Burstin's Sefer (פרק י' סעיף ג',ד'), and is cogently argued in the aforementioned essay by Dr. Yisrael Ziderman (בספר "תחומין" תשמ"ח שם), and has been presented to contemporary Poskim and Rabbonim. It should be noted, though, that even if one has the true Techeiles, it is questionable if one should use it today, as pointed out by the Beis HaLeivi, quoted by Rabbi Leiner ( עמוד י"גהקדמה למאמר עין התכלת,); there is, however, some question as to exactly what the Beis HaLeivi meant, as documented by HaRav Hershel Schachter, citing HaRav Yosef Dov Soloveitchik (ספר נפש הרב, בעניני המסורה, עמוד נ"ג-נ"ד, ובהערה 26 שם).
If one does use Techeiles in his Tzitzis, there is then a question as to how many of the threads have to be dyed that color. According to Rashi in Menachos (דף ל"ח. בד"ה התכלת) and elsewhere, two of the four threads (on each corner of the garment) should be Techeiles; since the threads are doubled over, there are four Techeiles threads and four white ones on every corner. This is also the position of Tosafos in Menachos (שם בד"ה התכלת) and elsewhere, as well as of the Rosh in both the Hilchos Tzitzis section of his Halachos Ketanos (סימן ו') and in a Teshuvah (שו"ת הרא"ש כלל ב' סימן ט',י"ב), and several other Rishonim. The Rambam, however (פרק א' מהל' ציצית הלכה ו'), writes that only one of the eight threads should be made of Techeiles, which means, as he explains in a Teshuvah (שו"ת הרמב"ם מהדורת בלאו חלק ב' סימן רפ"ו), that one half of one of the original four threads is dyed Techeiles. The Ohr Somayach, commenting on the Rambam (הל' ציצית שם), notes that this view is based on the Yerushalmi in Kilayim (פרק ט' הלכה א', דף מ.); according to Rabbi Leiner (מאמר פתיל תכלת, עמוד קנ"ד), this is also the view of Rav Yosef Karo, in his Sefer Maggid Meisharim on Chumash, and of the Ari Zal. Finally, the Raavad (להל' ציצית שם השגת הראב"ד) holds that two of the eight threads must be made of Techeiles, meaning that one of the four, before the doubling over, is to be dyed Techeiles; this view is quoted as well by the Aruch (ערך תכלת), and appears to be the position of the Meiri in Yevamos (בית הבחירה לדף ד: שם בסוף ד"ה והדר). It is the first view, however, that of Rashi (שם), that seems to be most widely quoted; it is the one codified by the Tur (או"ח סימן י"א), and is also accepted by the Taz (שם ס"ק י"ג) and the Mishnah Berurah (שם ס"ק נ"ח), as well as by the Vilna Gaon later (ביאור הגר"א לסימן י"ב שם בד"ה ולר"ת).
There is also some question as to the winding of the strings (כריכות) and the tying of the knots (קשרים) when using Techeiles in Tzitzis, and hence there are different practices. The Gemara in Menachos (דף ל"ט.) speaks of winding a "חוליא;" Rashi (שם בד"ה אלא) defines this as a unit which has one string wrapped around the others three times. The Gemara (שם) then says that one should do this no less than three and no more than seven times, starting and ending the windings by wrapping a white string around the others; there is no indication, however, as to how to do the rest of the windings, whether with white or with Techeiles, or some combination, nor are we told how many times to wind and how many knots to tie. Rashi in Menachos (שם בד"ה מין כנף) presents one way of tying Tzitzis with Techeiles, Tosafos (שם בד"ה לא) offers two other approaches, the Rambam, in his aforementioned Teshuvah (שם), as well as in his Mishneh Torah (הל' ציצית שם הלכה ז'), suggests yet another, the Raavad (השגת הראב"ד שם) quotes another approach from the Geonim, and there are still other opinions about all this, leading to many different possibilities. It should be noted, though, that it is clear from the Gemara (שם) that all one really needs is to have one wound חוליא (a triplet), and only one knot, and this idea is codified in the Shulchan Aruch (שם סימן י"א סעיף י"ג) and explained by the Mishnah Berurah (שם ס"ק ס"ו); the other approaches are merely the preferable ones according to the different authorities.