A subring with unity is a subring which includes, not just any unity, but the same unity as the parent ring. Of course this is the only sort of subring we talk about if we define rings to have a unity by default. Every ring has a unique smallest subring with unity, isomorphic to either the IntegerNumbers or some ModularArithmetic.
Subrings closed under multiplication by arbitrary ring elements are called ideals. Clearly the only ideal which contains unity is the ring itself. These play exactly the same role that normal SubGroups do for groups, being precisely the possible kernels of ring HomoMorphisms.