A special case of what you are asking for are properties that are [Morita invariant](http://en.wikipedia.org/wiki/Morita_equivalence#Properties_preserved_by_equivalence). I just copy the list from the Wikipedia article:

    simple
    semisimple -- [see Fred Rohrer's answer]
    von Neumann regular -- [see Fred Rohrer's answer]
    right (or left) Noetherian -- [see user46855's answer]
    right (or left) Artinian
    right (or left) self-injective
    quasi-Frobenius -- [see user46855's answer]
    prime
    right (or left) primitive
    semiprime
    semiprimitive
    right (or left) (semi-)hereditary -- [see user46855's answer]
    right (or left) nonsingular
    right (or left) coherent -- [see user46855's answer]
    semiprimary
    right (or left) perfect
    semiperfect
    semilocal

There are more in Lam's book *Lectures on Modules and Rings* (§18 in particular), e.g. "finite uniform dimension" or even "finite cardinality". Of course it is not said that the description in terms of the module category is always as beautiful as in some of the specific answers; sometimes it is a nice exercise to search for a "concrete" and "nice" description in terms of modules.

I take the opportunity to advertise my question [MO/124856](http://mathoverflow.net/questions/124856/) which asks whether there is such a description for being classical.