When Hashem tells Moshe to inform Bnai Yisrael that He plans to take them out of Egypt, He uses four expressions in order to fully describe His precise intentions (שמות ו':ו'-ז'). The third of these expressions includes the word "וגאלתי," "and I shall redeem" (שם פסוק ו'). Rabbeinu Bechaya, in his commentary on the Torah (שם), states that this promise is an allusion to the splitting of the Red Sea, since only after that happened were the people truly considered redeemed (גאולים). He explains (שם) that just as a servant who has had his life embittered by a difficult master who is now forced to free him will still be frightened that this master will pursue him, and he will thus not feel completely redeemed until he hears that this master has died, so too Bnai Yisrael, even after being freed by Paroh, still lived with the fear that the Egyptians would pursue them and try to bring them back into slavery until the time of the splitting of the Red Sea, when they realized that with the death of the Egyptians, this was no longer a possibility. Only at that time did they appreciate the fact that they were truly redeemed.
Rabbeinu Bechaya (שם) then adds that it is for this reason that we do not find any form of the word גאולה, redemption, mentioned in connection with Yetzias Mitzrayim at all until after the destruction of the Egyptians, because only then were Bnai Yisrael considered completely redeemed and saved. It is noteworthy that in the song "Az Yashir," sung by Bnai Yisrael after the splitting of the Red Sea, Hashem is praised for specifically having redeemed the nation (שם ט"ו:י"ג). The Ramban, in his introductory remarks to Sefer Shemos (הקדמה לספר ואלה שמות, נדפס לפעמים בסוף ספר בראשית), takes this idea a step further, stating that Bnai Yisrael were not considered redeemed until they had reached the spiritual level achieved by the Avos, upon whose tents Hashem's Shechinah rested; this happened when the Mishkan was completed and Hashem dwelled therein, and this is why the details of the construction of the Mishkan are included in this Sefer of the Chumash, whose dominant theme is redemption. The term Geulah (גאולה), then, refers to a full and complete redemption, in both the physical and the spiritual sense; in common usage today, it refers to Yetzias Mitzrayim, and all that was part of that experience, when considering the past, and to the coming redemption in the days of Moshiach, and all that that will include, when looking towards the future.
The Gemara in Berachos (דף ד: ודף ט:) refers to a Beracha which is nicknamed "גאולה," the Beracha of Geulah, that is recited prior to saying the Shemoneh Esrei (תפילה) both in the morning and in the evening; the Gemara in Pesachim (דף קי"ז: ועיין שם ברשב"ם בד"ה קרית) states that this Beracha concludes with the words "גאל ישראל," indicating that Hashem is the redeemer of Klal Yisrael. The Mishnah in Berachos (דף י"א.) speaks of certain Berachos which surround the twice daily recitation of Kerias Shema, with two preceding it and one following it in the morning, and two preceding it and two following it in the evening; the Rambam (פרק א' מהל' קריאת שמע הלכה ה',ו') spells out these Berachos more clearly. The Bartenura, commenting on that Mishnah (שם פרק א' משנה ד' בד"ה ואחת), indicates first that the Beracha which follows Kerias Shema in the morning begins with the words "אמת ויציב," a fact which is hinted at by another Mishnah in Berachos (דף י"ג.), and he then indicates (שם בד"ה ושתים) that the Beracha which follows Kerias Shema in the evening begins with the words "אמת ואמונה," and the Rambam (שם הלכה ו') rules accordingly. It is clear from the Tur (אורח חיים סימן ס"ו) that it is this Beracha recited in the morning beginning with the words "אמת ויציב" that ends with the words "גאל ישראל," and the Tur later (שם סימן רל"ו) asserts that the Beracha recited in the evening beginning with the words "אמת ואמונה" also ends with the words "גאל ישראל." The Beracha of Geulah, then, is the one Beracha which follows Kerias Shema in the morning, and one of the two Berachos which follow Kerias Shema in the evening.
The Gemara elsewhere in Berachos (דף י"ב.) briefly presents the reason for the difference between the text of the Beracha in the morning as opposed to its text in the evening, connecting it to a Posuk in Tehillim (צ"ב:ג'). Rashi (שם בד"ה שנא') elaborates on this, explaining that the morning Beracha of אמת ויציב refers to the kindness which Hashem displayed by taking our ancestors out of Egypt and splitting the Red Sea for them, while the evening Beracha of אמת ואמונה refers as well to the redemption of the future, expressing the hope that Hashem will fulfill His promise and redeem us from our enemies and oppressors in the (near) future. Tosafos (שם בד"ה להגיד) explains this distinction similarly, stressing that the theme of חסד, kindness is appropriate for the morning Beracha, as hinted at in the aforementioned Posuk in Tehillim (שם), because it focuses on Hashem's past kindness, while the theme of אמונה, faith, is appropriate for the evening Beracha because it focuses on the future. Tosafos (שם) then adds that אמונה is an appropriate theme for the evening because we focus then on the fact that our souls are given to Hashem for the night and we have faith that He will return them to us. The Yerushalmi in Berachos (פרק א' הלכה ו', דף י"א:) describes other themes that are included in this morning Beracha; the Aruch HaShulchan (או"ח סימן ס"ו סעיף י"ז) and the Mishnah Berurah (שם ס"ק נ"ג) write that those themes should be included in the evening Beracha as well. The Gemara later in Berachos (דף כ"א.) cites a view that the recitation of the Beracha of אמת ויציב is required MideOraisa, and Tosafos (שם בד"ה ספק) stresses the importance of mentioning the theme of Yetzias Mitzrayim as a central motif of this Beracha; the Tur (או"ח שם) quotes all of these themes regarding the Berachos of אמת ויציב and אמת ואמונה.
The Midrash in Shemos Rabbah (פרשה כ"ב סימן ד') states that one must refer to Yetzias Mitzrayim in the Beracha of ביציו תמא, and if one does not, he must repeat the entire Beracha; the centrality of Yetzias Mitzrayim is then documented there. The Perishah, commenting on the Tur (שם אות י"ג), notes that even in the evening, when the Beracha focuses primarily on the future, as explained above, the end of the Beracha makes reference to the Geulah from Egypt because the conclusion of the Beracha is גאל ישראל"," in the past tense, referring to the past Geulah, as the Tur himself writes elsewhere (שם סימן רל"ו); the Bach (שם בד"ה וחותם) writes that even this Beracha was instituted principally to focus on the past Geulah, and we simply declare our faith that Hashem will redeem us in the future as He did in the past, a point echoed by the Perishah there (שם אות ה'). The Gemara in Berachos (דף י"ב.) states that one who fails to say ביציו תמא at Shacharis and אמת ואמונה at Maariv has not properly fulfilled his daily obligation, and the Rambam (שם הלכה ז') and the Shulchan Aruch (שם סימן ס"ו סעיף י') rule accordingly; the Magen Avraham (שם ס"ק י"ד) and others note that this person has fulfilled his obligation to recite Kerias Shema, but, as the Aruch HaShulchan (שם) and the Mishnah Berurah (שם) cited above state, he has not done so properly, as already noted by the Kessef Mishneh, commenting on the aforementioned Rambam (שם), who also states that the Beracha of אמת ויציב hints as well to the future Geulah.
The Gemara earlier in Berachos (דף ט:) states that one must juxtapose this Beracha of Geulah (concluding with לארשי לאג) with the beginning of the Shemoneh Esrei; this is known as הליפתל הלואג תכימס, juxtaposing this Beracha to the main body of Tefillah, which is the Shemoneh Esrei, and the Gemara (שם) speaks of the reward given to one who does this. The Rambam (פרק ז' מהל' תפילה הלכה י"ז) writes that one should stand up (to say the Shemoneh Esrei) immediately after reciting the Beracha of לארשי לאג; the aforementioned Tur (שם) writes as well that one should begin the Shemoneh Esrei right away, with no interruption after the Beracha of לארשי לאג, except for under certain extenuating circumstances. The Beis Yosef (שם בד"ה ומתחיל) adds that this applies regardless of when one is davening, and the Kaf HaChaim (שם אות ל"ג) writes that this is the view of the Ari Zal as well, based on Kabballistic sources. The Tur emphasizes again later (שם סימן קי"א) that there should be no interruption whatsoever between גאולה and תפילה, although one may answer אמן to the Beracha of לארשי לאג, because this is not considered an interruption; the Beis Yosef writes, however (שם בד"ה אבל אמן), that the practice is not to say אמן in between גאולה and תפילה. The Shulchan Aruch (שם סימן ס"ו סעיף ח') rules that one must juxtapose גאולה and תפילה and thus should not interrupt with anything after the Beracha of לארשי לאג, not even a silent pause, as pointed out by the Mishnah Berurah (שם ס"ק ל"ח), except in an emergency situation, as noted as well by the Mishnah Berurah (שם ס"ק ל"ט), and amplified by the Aruch HaShulchan (שם סעיף ט"ז, י"ז). The Shulchan Aruch later (שם סימן קי"א סעיף א') repeats that one must juxtapose גאולה and תפילה, adding that one may not interrupt between them at all, even to answer אמן to the Beracha of לארשי לאג or to recite any Posuk, with the exception of the Posuk of "ה' שפתי תפתח ופי יגיד תהלתך" (תהלים נ"א:י"ז), which the Gemara in Berachos (דף ד: ודף ט:) indicates is considered part of the Shemoneh Esrei; the Ramo (שם) rules, though, that one may answer אמן to the Beracha of לארשי לאג. The Mishnah Berurah (שם סימן ס"ו ס"ק ל"ה) writes that in order to satisfy both opinions, one should try to conclude his own recitation of the Beracha of לארשי לאג together with that of the Chazzan so that he will not have to answer אמן at all, or he should begin his Shemoneh Esrei slightly before the Chazzan says לארשי לאג so that he will not have to answer אמן; the former suggestion may be found in the Magen Avraham (שם ס"ק י"א) as well.
The Gemara in Berachos (דף ט:) notes that it is proper to have הליפתל הלואג תכימס at Maariv as well, an idea presented as the subject of a dispute by a previous Gemara there (שם דף ד:, ועיין שם בתוד"ה דאמר). The Rambam (הל' תפילה שם הלכה י"ח) rules that it should be done, as does the Shulchan Aruch (שם סימן רל"ו סעיף ב'), although it appears from the Ramo (שם), as explained by the Mishnah Berurah (שם ס"ק ט'), that there are certain leniencies which apply to the requirement of הליפתל הלואג תכימס at Maariv that do not apply at Shacharis (ועיין עוד שם בשער הציון אות ד'). In general, though, one should not have any interruptions between גאולה and תפילה at Maariv either, as noted earlier by the Mishnah Berurah (שם סימן קי"א ס"ק ד') and by the Kaf HaChaim (שם אות ג'). It is noteworthy, though, that the Rosh in Berachos (פרק א' סוף סימן א'), among others, quotes from Rav Hai Gaon that one may forego הליפתל הלואג תכימס at Maariv in order to be able to say the Shemoneh Esrei with a Minyan, meaning that if one comes late to Shul and the Tzibbur is starting the Shemoneh Esrei of Maariv, one should join them despite not yet having said Kerias Shema and its Berachos, which he must then make up later, obviously without having הליפתל הלואג תכימס. The Rashba (שו"ת הרשב"א חלק א' סימן רל"ו) writes that this is because Maariv is itself (originally) an optional Tefillah, so these laws are less strict; the Shulchan Aruch (שם סימן רל"ו סעיף ג') rules accordingly. At Shacharis, however, one must have הליפתל הלואג תכימס, even if it means davening without a Minyan, as the Rashba (שם) states; the Shulchan Aruch (שם סימן קי"א סעיף ג') again rules accordingly.
Rashi in Berachos (דף ד: בד"ה זה) refers to a Yerushalmi in Berachos (פרק א' הלכה א', דף ו.) which rebukes one who fails to connect גאולה and תפילה, and praises one who does so; Rabbeinu Yonah in Berachos (דף ב: בדפי הרי"ף בד"ה איזהו) elaborates on this, saying that when one does this, one demonstrates that he understands that because Hashem redeemed us, we must serve Him through Tefillah, and that just as our ancestors trusted Hashem and He saved them, so too we trust that Hashem will answer our Tefillos, and that this trust in Hashem is a foundation of our faith. The Yerushalmi (שם) also derives this whole idea of הליפתל הלואג תכימס from the close proximity of two Pesukim in Tehillim (י"ט:ט"ו, כ':ב'), and the Tur (שם סימן קי"א) explains that the first Posuk (יהיו לרצון אמרי פי והגיון לבי לפניך ה' צורי וגואלי) refers to Geulah, while the other one (יענך ה' ביום צרה...) refers to Tefillah; since these two themes are juxtaposed in Tehillim (שם ושם), they should be juxtaposed when we daven. The Bach, commenting on the Tur (שם בד"ה ויסמוך), quotes from the Maharshal that the idea of הליפתל הלואג תכימס can actually be learned from the former Posuk (יהיו לרצון...) alone since it includes both themes by itself, but the Perishah (שם אות ב') challenges this view, noting that the theme of Tefillah comes first in that Posuk (שם), if it is indeed referred to there at all, which he questions, and he includes that the second Posuk (שם) is therefore necessary, as does the Taz (שם ס"ק א') who similarly rejects the view of the Maharshal and explains the derivation from the two Pesukim clearly. The Bach (שם) adds, though, that this entire derivation is in the category of an אסמכתא, a hint from the Pesukim, and the idea of הליפתל הלואג תכימס is really simply an institution by the Rabbanan; the Beis Yosef (שם בד"ה וכתוב) says this as well.
The Beis Yosef (שם) does, however, quote from the Hagahos HaAsheri in Berachos (פרק א' סימן י') that since the second Posuk in Tehillim cited above (שם כ':ב'), referring to Tefillah, speaks of a יום צרה, a day of distress, הליפתל הלואג תכימס is not necessary on Shabbos because Shabbos is not a day of distress, and therefore the derivation from that Posuk (שם) described above is inapplicable. The Beis Yosef (שם) himself rejects this opinion, however, saying, as explained above, that the Posuk (שם) is intended merely as a hint, and הליפתל הלואג תכימס would be required in any case, and adding that Tefillah on Shabbos simply takes the place of Tefillah on a weekday, which is an עת צרה, a time of distress; the Tiferes Shemuel, commenting on the Hagahos HaAsheri in Berachos (שם אות כ"ד) also rejects this view. The Ramo, however, in his Darkei Moshe on the Tur (שם אות א'), quotes that the Kol Bo (סימן ל"ה) accepts this view, which is rooted in the Ohr Zarua (חלק א' סימן י"ד, וחלק ב' סימן מ"ב) in the name of Rabbeinu Tam, as does the Maharil, whom he quotes as stating that on Yom Tov, הליפתל הלואג תכימס is nevertheless required because that day of the week is in general a יום צרה, while Shabbos never is. The Ramo (שם) thus concludes that one should not interrupt between גאולה and תפילה even on Shabbos for no real reason, but for a good reason, there is room to be lenient. The Pri Chodosh (או"ח סימן קי"א ס"ק א') writes that it appears from the Rosh in Taanis (פרק א' סימן ב') that הליפתל הלואג תכימס is required on Yom Tov; the Shaagas Aryeh (שו"ת שאגת אריה סימן ט"ז) also quotes this view of the Rosh (שם), and makes a strong case that הליפתל הלואג תכימס is required on Shabbos as well, although he does defend the position that it is not required on either Shabbos or Yom Tov. It would appear from the Rashba (שו"ת הרשב"א המיוחסות להרמב"ן סוף סימן קפ"ח) as well as from the Raavad (פרק ג' מהל' תפילה הלכה ז') that הליפתל הלואג תכימס is indeed required on Shabbos; Rav Ovadyah Yosef (ספר ילקוט יוסף חלק א', הלכות תפילה הערה נ"ה) quotes many others who hold this way as well. In the Shulchan Aruch (שם סעיף א'), the Ramo quotes the view that הליפתל הלואג תכימס is not required on Shabbos, but he concludes that one should be strict then too unless there is a great need; the Mishnah Berurah (שם ס"ק ט' ועיין שם בביאור הלכה בד"ה וטוב) explains that this means that one may answer אמן יהא שמיה רבא, Kedushah, and Borechu between גאולה and תפילה on Shabbos, even at Shacharis, something which one may not do on a weekday, as stated earlier in the Shulchan Aruch (שם סימן ס"ו סעיף ט').