Given a field an affine algebra is any finitely generated noetherian associative commutative unital -algebra without nilpotent elements. This term is limited to a small community of algebraists and may bring confusion if quoted outside of that context. The name is from the fact that the affine algebras are the coordinate rings of affine varieties; equivalently, an affine -variety is precisely the maximal spectrum of an affine -algebra.
Some people drop the condition of commutativity and talk about noncommutative affine algebras. This is even less standard as in the noncommutative context there is a further hesitation to drop noetherianess as well.
Yet another confusion may arise when people informally say affine algebra for an affine Lie algebra; of course, the latter, full, term is preferred (or “affine Kac-Moody algebra”).