One of the special gifts given to the kohanim was a portion of every kosher animal slaughtered for food. Specifically, they received the cheeks, the right foreleg and the abomasum, which is one of the stomachs. This only applies to domesticated animals (such as cows and sheep), not to wild game (like deer).
The Talmud in Chulin (134b) explains that this gift was given to the kohanim in the merit of Pinchas, who acted zealously for God (in Numbers chapter 25). The original kohanim were only Aaron and his sons; any grandsons who had already been born remained Leviim. (Pinchas was one of these grandsons.) Not only did Pinchas' actions earn him a “promotion” to kohein, it earned his descendants (as well as all the other kohanim) this special portion. Such is the power of making a kiddush Hashem (sanctification of God's Name)!
This gift could be given to any kohein, even a woman. This is true even if she married a non-kohein, which disqualifies her from eating terumah. Furthermore, even her non-kohein husband is allowed to “ride on her coattails” and eat these portions because of his wife’s priestly descent. It was unclear whether or not Leviim were obligated in giving this gift, so the practice arose for kohanim not to take them from Leviim. (If they did take them, they were not returned.)
There is some discussion among the authorities as to whether or not this mitzvah applies today. Since we generally buy our meat from butchers nowadays, the average person need not be concerned with the matter as it is strictly between the butcher and the kohein.
In the Talmud, this mitzvah is discussed in tractate Chullin starting on page 130a. It is codified in the Mishneh Torah in the ninth chapter of Hilchos Bikkurim. This mitzvah is #143 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos and #51 of the 77 positive mitzvos that can be fulfilled today in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim. (The Chofetz Chaim acknowledges the difference of opinion regarding this mitzvah and rules that our practice is to be lenient, though exceptional people act stringently.)