The afikoman is the piece of the middle matzah at the Pesach Seder which ends up being pulverised under the carpet, in the DVD player, or in whatever other brilliant hiding place Dad thought of this year.
Its etymology is unclear. It appears to come from Greek: one theory is that it is derived from "epi," meaning "around" and "komao," "to revel." Jastrow's Talmudic Dictionary quotes Plutarch using the Latin version "commessatum," to mean "to the after-dinner entertainment."
On this reading, the enigmatic statement of the Mishnah, "Ein maftirin achar ha-Pesach afikoman"(Pesachim, 10:8) means, "After the Seder meal is finished, one shouldn't finish off by saying ‘Let's go to the after-dinner entertainment.'" The entertainment that evening is to carry on talking about the Exodus from Egypt late into the night.
According to a variant of this view, afikoman was a Greek practice of going from house to house after dinner for a snack. The rabbis wanted to prevent this happening on Seder night, so that people would celebrate the whole occasion with the same family group. The afikoman matzah, which is the last food we taste at the meal is instead of the afikoman practice of roaming the streets after dinner.