The Torah tells us that on the second day of Pesach (the 16th of Nissan), a special offering had to be brought to Hashem consisting of the grain from the first harvest of that season (ויקרא כ"ג:י-י"א). Since the Torah specifies (שם פסוק י') that the amount of this grain brought was to be an Omer's worth, meaning, as the Torah indicates earlier (שמות ט"ז:ל"ו), one tenth of an Eiphah, the equivalent of about two quarts, this offering was known simply as the Korban Omer. The Torah elsewhere (ויקרא ב:י"ד) indicates, as interpreted by the Gemara in Menachos (דף ס"ח:) based on another Posuk (שמות ט:ל"א), that this first grain offering consisted specifically of barley; the barley was roasted and then ground into a kind of meal, as the Gemara earlier (שם דף ס"ו:) states. The Rambam (פרק ז' מהל' תמידין ומוספין הלכות י"א, י"ב) clearly details each step of this Korban from the preparations for the harvesting of the barley through the actual offering.
The Torah then states (ויקרא כ"ג:י"ד) that until this Korban Omer is brought, it is forbidden for one to eat bread or grain products. The Mishnah in Menachos (דף ע.), after specifying that this injunction applies to the חמשת מיני דגן, five species of grain, namely, wheat, barley, spelt, oats, and rye, explains that the prohibition is to eat any of these grains (or their derivatives) which had grown, or, more literally, took root, during the past year, since the Korban Omer was brought on the previous Pesach, until the current Korban Omer is brought. Any such grain which begins to take root after Pesach is called "Chodosh," meaning "new" by this Mishnah, and it is forbidden to eat Chodosh or products made from Chodosh until the Korban Omer is brought on the next Pesach. The Rambam (פרק י' מהל' מאכלות אסורות הלכה ד') and the Shulchan Aruch (יורה דעה סימן רצ"ג סעיף ג') rule accordingly.
An earlier Mishnah in Menachos (דף ס"ח.) states that in the absence of the Beis HaMikdash, when no Korban Omer is brought at all, Chodosh becomes permissible to eat only following the day on which it would have been brought, that is, the 16th of Nissan. One may eat Chodosh, then, starting on the evening of the 17th of Nissan, and, based on the Gemara's conclusion (עמוד ב' שם), on the evening of the 18th of Nissan if one lives outside of Eretz Yisrael, where an extra day of Yom Tov is observed because of a doubt (at one time) as to the true calendar date. The Rambam (שם הלכה ב') and the Shulchan Aruch (שם סימן א') again rule accordingly.
It should be noted that there is a dispute as to how long it takes for these grains to take root after they're planted. The Shach (שם ס"ק ב') quotes from the Terumas HaDeshen (שו"ת תרומת הדשן שאלה קצ"א) that it takes three days, based on an opinion in the Gemara in Pesachim (דף נ"ה.), but Rabbi Akiva Eiger (חידושי רע"ק שם בד"ה היינו), among others, questions this, saying that that opinion is not the accepted one, and that it rather takes two weeks. The Shach himself, in his Nekudos HaKessef (שם בד"ה כדמוכח) raises this question, noting that the Gemara in Yevamos (דף פ"ג.) seems to rule clearly that it takes two weeks. The Vilna Gaon (ביאור הגר"א שם סוף ס"ק ב') brings this up as well and tries to reconcile the problem, but leaves the matter in doubt. The Aruch HaShulchan (שם סעיפים ז-ט) summarizes the different positions, but concludes that the view of the Terumas HaDeshen that it takes three days is correct when discussing, as we are, plants, as opposed to trees. This dispute would obviously have great bearing on any grains planted in the springtime just before Pesach in determining whether or not they'd be labeled as Chodosh.
In formulating this Mitzvah not to eat Chodosh, the Torah (שם) specifies that it is to be observed בכל משבתיכם"," "wherever you dwell." This would imply that this Mitzvah is not restricted to Eretz Yisrael or to its produce. Indeed, the Mishnah in Orlah (פרק ג' משנה ט') states plainly that the prohibition to eat Chodosh applies everywhere according to the Torah. The Mishnah in Kiddushin (דף ל"ו:-ל"ז.), however, presents a dispute about this, implying that the majority of opinions hold that Chodosh in fact applies only in Eretz Yisrael. In the ensuing discussion, the Gemara (שם) suggests that the phrase בכל משבתיכם may be understood to teach not that the Mitzvah applies outside of Eretz Yisrael, but that even in Eretz Yisrael it was not to be operative until the land had indeed become a dwelling place, that is, after the entire conquest and division of the land. The Yerushalmi in Kiddushin (פרק א' הלכה ח' דף כ"ב.) suggests that although produce grown outside of Eretz Yisrael is not subject to the laws of Chodosh, the phrase בכל משבתיכם teaches that Chodosh produce from Eretz Yisrael which is brought outside the land may still not be eaten. The aforementioned Gemara in Menachos (דף ס"ח:) presents this dispute somewhat differently: some Amoraim learn that the Mitzvah of Chodosh applies outside of Eretz Yisrael on a Torah level, while others hold that the Mitzvah is MideRabbanan anywhere outside the land; either way, though, the Mitzvah applies everywhere.
The Rambam cited above (שם הכלה ב') rules clearly that the Mitzvah of Chodosh applies on a Torah level everywhere, as do the Rif in Kiddushin (דף ט"ו: בדפיו) and the Rosh there (פרק א' סימן ס"ב) and others. Elsewhere, however, the Rosh (שו"ת הרא"ש כלל ב' סימן א') quotes some Rishonim who hold that there is a doubt as to whether this Mitzvah applies outside of Eretz Yisrael, and others who hold that it applies only MideRabbanan outside the land, and still others who hold that even MideRabbanan it applies only to the lands immediately neighboring Eretz Yisrael. The Aruch HaShulchan (יו"ד שם סעיפים ב'-ו') presents a synopsis and a discussion of all of these views, and the basis for their positions.
The Shulchan Aruch quoted above (יו"ד שם סעיף ב') rules that the Mitzvah of Chodosh applies both in Eretz Yisrael and outside the land, adding that it doesn't matter whether the particular field is owned by a Jew or by a non-Jew, the subject of a different dispute. The Ramo, however, (שם סעיף ג'), writes that because of certain doubts which generally prevail as to when most available grains actually grew, one may be lenient and disregard the problem of Chodosh, unless one is indeed sure when the grain grew. He then adds that even when it is proper to be stringent with this Mitzvah, one should not publicize this Halacha if people generally use Chodosh products, because it is better for people to err unintentionally than to err intentionally.
The long-standing practice in most communities, as already noted by the aforementioned Terumas HaDeshen (שם), has been to be lenient, permitting eating Chodosh products grown outside of Eretz Yisrael; many Poskim have attempted to defend this leniency. The Taz (שם ס"ק ד'), for example, tries to explain why we may be lenient even if it means following the minority view. The Bach, in his commentary to the Tur (יו"ד שם בד"ה כתיב), writes that in his country, the practice even among the Gedolei Torah and their students is to be lenient about this, and that it is not clear from the Gemara that Chodosh applies anywhere but in Eretz Yisrael. He thus concludes that no Torah authority should prohibit eating Chodosh, ruling against the accepted leniency, and only one who is known as an exceptionally pious person who is strict about other things should accept this stringency upon himself as a מדת חסידות, an act of extra piety.
It must be stressed, however, that the Vilna Gaon (ביאור הגר"א שם ס"ק ב') disagrees strongly and uncharacteristically attacks the lenient opinions sharply, writing forcefully that the sources indicate that Chodosh applies even outside of Eretz Yisrael. The Magen Avraham (אורח חיים סימן תפ"ט ס"ק י"ז), while defending the lenient position from a number of points of view, concludes that it is proper for one who wishes to be stricter to do so. The Mishnah Berurah (שם ס"ק מ"ה), after summarizing the different positions, likewise writes that while one shouldn't object to those who are lenient, one should personally try to take the stricter view and avoid eating Chodosh products even outside of Eretz Yisrael.