CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $pro(FinSet)$ of pro-objects in FinSet and finite sets sit $FinSet\hookrightarrow pro(FinSet)$ as finite discrete spaces. This is especially common when talking about profinite groups and related topics.
An internal group in the category of Stone spaces / profinite spaces and continuous maps is a profinite group.
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.
The primary meaning (as Stone space) is used in sources on profinite groups, for which see the entries Stone space, profinite group.
Discussion of homotopy theory of pro(finite) simplicial sets is in
G. Quick, Profinite homotopy theory, Documenta Mathematica, 13, (2008), 585–612, (Archiv 0803.4082)
Jacob Lurie, Profinite spaces (pdf)