After the Seder meal, we eat a kezayit (about a third of a machine matza) of the broken matza that was set aside at the beginning of the Seder. This matza is called the afikoman. After eating the afikoman, we do not eat anything else until we go to sleep, so that the taste of the matza lingers in our mouths (SA 477:1, 478:1). If the afikoman set aside at the beginning is not big enough to give each participant a kezayit, the Seder leader gives each participant a small piece of the afikoman together with additional shemura matza to constitute a kezayit. If there is not enough afikoman to give even a small piece to each participant, the leader may give out shemura matza to be eaten as the afikoman. Likewise, if the afikoman was lost, one may use a different piece of shemura matza instead (Rema 477:2).
One may drink water after eating the afikomen, because the prohibition of eating after the afikoman is to ensure that its taste stays in the mouth, and since water has no taste, one may drink it after eating the afikoman (SA 478:1, MB 2 ad loc.).
The word “afikoman” originally meant “dessert”. On the night of the 15th of Nisan, the last thing one eats must be the meat of the Paschal sacrifice, so that its taste lingers with us, as the Mishna states: “after the Paschal sacrifice, we do not conclude with dessert (afikoman)” (Pesaḥim 119b).
Since we can no longer offer the korban Pesaḥ, the Sages ordained eating matza at the end of the Seder to commemorate the korban Pesaḥ. So that the taste of this matza remains in our mouths, nothing may be eaten after it. Since this matza is the final course of the Seder, it is in essence dessert. Therefore we call it “afikoman.”
Since the afikoman commemorates the korban Pesaḥ, and just as the Paschal sacrifice was eaten “while satisfied” (“al ha-sova”), so too the afikoman must be eaten while satisfied (SA 477:1). “Al ha-sova” means that one is already satiated but still wants to eat more. However, if one is so full that his appetite is gone, he does not fulfill the mitzva in the ideal manner, since he would prefer not to eat any more. If one is so stuffed that food is repugnant to him, but he nonetheless forces himself to eat the afikoman, he is engaged in “akhila gasa” (gross overconsumption), which is not considered eating; one does not fulfill the mitzva of eating the afikoman by eating in this manner (MB 476:6, Kaf Ha-ḥayim 17 ad loc.).
The afikoman must be eaten in one place, because it commemorates the Paschal sacrifice, which had to be eaten in one place, as it is stated (Shemot 12:46): “It should be eaten in one house” (Rema 478:1, MB 4 ad loc.).