A roleclaim (or just "claim") is a statement of what your role is.
Players may claim if they feel they have results they need to give to the Town, or if they are ordered to do so by the Town (i.e. in massclaim), or in order to save themselves from being eliminated (towns generally don't want to eliminate one of their power roles).
Note that due to the possibility of it becoming a breaking strategy that is outside the spirit of the game, most of the time players are disallowed from directly quoting their Role PM in a claim. Paraphrasing it is okay, though.
On some sites other than mafiascum.net, claiming in any context is not allowed. This is usually considered a sign of a weak setup that can be broken (by a mass nameclaim, for instance).