A special case of what you are asking for are properties that are Morita invariant. 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 which asks whether there is such a description for being classical.

A special case of what you are asking for are properties that are Morita invariant. 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 which asks whether there is such a description for being classical.

A special case of what you are asking for are properties that are Morita invariant. I just copy the list from the Wikipedia article:

simple, semisimple
von Neumann regular
right (or left) Noetherian, right (or left) Artinian
right (or left) self-injective
quasi-Frobenius
prime, right (or left) primitive, semiprime, semiprimitive
right (or left) (semi-)hereditary
right (or left) nonsingular
right (or left) coherent
semiprimary, right (or left) perfect, semiperfect
semilocal

There are more in Lam's book Lectures on Modules and Rings (section 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 which asks whether there is such a description for being classical.

A special case of what you are asking for are properties that are Morita invariant. 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 which asks whether there is such a description for being classical.