Yoshon, literally "old", refers to the "old grain" that took root before the previous 16th of Nissan and is permitted to eat, according to Torah law. Chodosh, literally "new", refers to the "new grain" that took root on or after the previous 16th of Nissan and, according to Torah law, is forbidden to eat.
(פסוק יד) ולחם וקלי וכרמל לא תאכלו, עד עצם היום הזה:עד הביאכם, את קרבן א-להיכם: חקת עולם לדרתיכם, בכל משבתיכם. [ס]
"And bread, and roasted (grain), and roasted (grain) sheaves you shall not eat until the middle of this day, until you will bring the offering of your G-d, an eternal law for your generations, in all of your dwelling places." (Vayikra: 23, 14)
The above is the source for the Torah prohibition for eating chodosh. Chodosh refers to any of the five types of grain (barley, rye, oat, wheat, and spelt) that took root after the 16th of the month of Nissan.
Definition of "taking root": In order to determine which grains are permitted and which are forbidden, it is necessary to know the halachic (Jewish legal) period of time the grain takes to have taken root (from the time of planting). Today, we follow the ruling of Rabbi Yossi - The grain takes two weeks to take root. The RaMBa"M (Maimonides) rules according to Rabbi Yossi - the grain takes two weeks to be considered to have taken root. (Laws of Ma'aser Sheni and Neta Rivai: Ch. 9, Mishna 8) The other halachic ruling (the one we do not hold by today) is by Rabbi Yehuda - grain takes root after three days - Pesachim 55a.
There is much discussion among the rabbinic commentators precisely when the grain must take root in order to be permitted after the omer offering on the 16th of Nissan. To avoid getting involved in the specifics, it is best (and in my opinion, simplest) to follow the ruling of the Shulchan Aruch (Yorah Deah: Siman 293, Sif 3): "Grain that has not taken root before the 16th of Nissan is forbidden [for use] until the next [year's] Omer [offering]." Therefore, in order to halachically ensure that the grain will be yoshon, it must be sowed by the first day of Nissan. As the Jewish calendar is a lunar calendar, it is not consistent with the solar calendar. For example, the grain must be sowed by (the day of) March 20 in 2007 or April 6 in 2008 to be able to be used prior to the 17th of Nissan of the following year (April 22, 2008 and April 11, 2009, respectively - outside of Israel, one would have to wait one more day outside the Land of Israel).
Above: Spring Wheat - Chodosh or Yoshon in its first year?
The Torah prohibition which does not allow the eating of chodosh applies in Israel. There is a rabbinic dispute as to whether chodosh applies to grains grown outside Eretz Yisroel (The Land of Israel).
CHODOSH AND GRAIN GROWN OUTSIDE THE LAND OF ISRAEL:
According to the Shulchan Aruch (Siman 293, Sif 2), "The prohibition of the chodosh applies both in the Land of Israel and outside the Land of Israel, whether [of grain] belonging to a Jew or [belonging to a] non-Jew." Other Torah greats who hold that the Torah prohibition of chodosh applies all over the world include the RaMBa"M (Laws of Forbidden Foods: Chapter 10, mishnah 2), RI"F, author of Sefer HaChinuch. The RI"F holds that the prohibition is a safek d'oraisa (possibly a Torah law that applies) outside Eretz Yisroel. See also maseches Menachos 70a. According to the RaMBa"M and the author of Sefer HaChinuch, if one eats of foods made from chodosh grain, he/she is liable for malkus.It is important to note that outside the Land of Israel, it is forbidden to eat chodosh grain (according to rabbinic law) until the 18th of Nissan - the fourth day of Passover, while inside the Land of Israel, one must wait until the 17th of Nissan (the third day of Passover) until all chadash becomes yashan.Others argue that grain grown outside of Eretz Yisroel (with the exception of nearby lands), the prohibition of chodosh does not apply. These rabbis include the ROS"H, Ba"CH, Magen Avraham, and Rabbi Akiva Eiger as well as Kiddushin 37a.To conclude, there is whom to rely upon even if one wishes to eat from grain grown, for example, in America that would otherwise be considered "chodosh". However, nowadays, there are many kosher products in America that are marked as "yoshon". In addition, from what I have seen, a majority of poskim have ruled that chodosh does apply to grain grown outside Eretz Yisroel.If one wishes to follow the Shulchan Aruch (and many others) and be careful to avoid eating from chodosh grain, no matter where it is grown, he/she should contact Rabbi Yosef Herman, an expert in the field of yoshon. Rabbi Herman provides booklets noting which products are permissible to the yoshon-eating public, when during the year they are permitted, as well as what restaurants are certified as yoshon-friendly. For Rabbi Herman's website, click here, his e-mail is: yherman@earthlink.net. Pass this information on to friends, enjoy, and bon appetite.
The OU has an informative article on yoshon.