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 shmura 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 shmura matza to be eaten as the afikoman. Likewise, if the afikoman was lost, one may use a different piece of shmura matza instead (Rema 477:2).
One may drink water after eating the afikoman. After all, eating after the afikoman is prohibited only so its taste stays in our mouths, and water has no taste (SA 478:1, MB 2 ad loc.).
The word “afikoman” originally meant “dessert.” On the night of the fifteenth 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)” (Pesaĥim 119b).
Since we can no longer offer the korban Pesaĥ, the Sages ordained eating matza at the end of the Seder to commemorate the korban Pesaĥ. 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.”
SA 477:1 explains that the afikoman commemorates the korban Pesaĥ, and just as the Paschal sacrifice was eaten “while satisfied” (“al ha-sova”), so too the afikoman must be eaten while satisfied. “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, this is called akhila gasa (gross overconsumption), which is not considered eating at all. One who does so does not fulfill the mitzva of eating the afikoman (MB 476:6, Kaf Ha-ĥayim 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.).