A fate is a brief statement on something a character has to do or an imperative during the course of a larp.
There are several different interpretations of the fateplay idea, but they usually involve an instruction that the player should be able to achieve easily, with the challenge being more about achieving it in an interesting an meaningful way. For example a character might be given the fate that "Before the second night of the larp, you shall challenge the king to a duel on the third morning. You will loose this duell." The duell is unavoidable, but the details of the challenge and the fighting leave plenty of space for dramatic improvisation.
Fates may be inter-connected into a "fateweb" where individual fates reveal their meaning as they are carried out. For example, the King in the above example might be fated to reveal his complicty in the murder of the previous King before the second day, thereby providing a motiviation for the character fated to challenge him to a duell.
Fates may be used to adopt whole narratives to larp - e.g. stories derived from mythology, theatre plays and the like. "Light-weight fates" may be used to give the players good ideas for what to do during the larp. There are also examples of larps where the first part of the larp is fated, and the fates are used to establish interesting conflicts, while the rest of the larp is open-ended.
History and further reading
Fateplay was probably "invented" in Norway ("Skjebne" or "Skjebnespill"). Some (old) texts on fateplay are gathered at: Amor Fati