A profinite set is a pro-object in FinSet. By Stone duality these are equivalent to Stone spaces and thus are often called profinite spaces. So these are compact Hausdorff totally disconnected topological spaces.
These are precisely the spaces which are small cofiltered limits of finite discrete spaces, and moreover (as a consequence of Stone duality) the category of Stone spaces is equivalent to the category of pro-objects in FinSet and finite sets sit as finite discrete spaces. This is especially common when talking about profinite groups and related topics.
Just as the term ‘space’ is used by some schools of algebraic topologists as a synonym for simplicial set, so ‘profinite space’ is sometimes used as meaning a ‘simplicial object in the category of compact and totally disconnected topological spaces’, i.e. in the other terminology a ‘simplicial profinite space’. This is further complicated by the question of whether or not pro(finite simplicial sets) and simplicial profinite spaces are the same thing.