Bikkurim were the first fruits to ripen on a tree of one of the special species of the land of Israel. (There are seven such species, singled out in Deut. 8:8: wheat, barley, grapes, figs, pomegranates, olives and dates. Of course, not all of them grow on trees.) These first fruits were one of the special portions given to the kohanim (priests). Those who lived near Jerusalem would deliver them fresh, but those who lived far away from the Temple would dry them for the journey.
The bikkurim were given to the kohanim on duty the week in which the owner delivered them. They had to be delivered in a vessel and, since they were terumah (a priestly gift), non-kohanim were not permitted to eat them. The laws of bikkurim only applied to the produce of Israel (including trans-Jordan) and Syria, not to produce from other lands. (Syria was a territory captured by Israel in the time of King David and was made subject to many laws that would otherwise be Israel-specific. Refer back to Mitzvah #84 or Talmud Gittin 8b.)
The underlying principle of this mitzvah is to recognize that all that we have comes from God. All too often, people remember to turn to Him in tough times but forget all about Him when things are going well. Here, as a person rejoices on the produce of his trees, he is to remember God and thank Him for the prosperity He has granted.
This mitzvah only applied at a time when the Temple was standing. It is the subject of the entire Mishnaic tractate of Bikkurim, but it is also discussed in the Talmud in Baba Basra 26b-27b. It is codified in the Mishneh Torah in Hilchos Bikkurim. The obligation to bring the first fruits is #125 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.