The assertion, claim, declaration, or statement of a party to an action, made in a pleading, setting out what he expects to prove.

In Plain English:

When the plaintiff and defendant file the complaint and answer, each of them tries to tell the facts of the case. The plaintiff will also explain why he is suing the defendant, and the defendant will explain his defense. These statements, until they are proven to be true or untrue in court, are called allegations.