There is no standardized definition. Using one of these terms always implies there is a ground field $k$, and (pre)varieties are $k$-schemes of finite type, possibly with certain restrictions (separated, reduced, irreducible...).
As far as I know, "prevariety" and "abstract variety" are no longer in use. "Prevariety" used to mean "possibly nonseparated" (at least in Mumford's Red Book, where prevarieties are in addition irreducible and reduced). "Abstract" stands for "not embedded in projective space".
Regrettably, some authors use the term "(algebraic) variety" without bothering to define it. In general, such carelessness is an indication that the objects in question are assumed separated and reduced (other cases not being worthy of consideration), or even that $k=\mathbb{C}$ (for the same reason).