A more abstract version of "matrix ring" is the endomorphism ring of a module. If you take your module to be a free module over R, then you get matrix rings, but there are plenty of other examples of modules that are worth thinking about. This is my go-to example when I need a ring that isn't necessarily commutative.
There was another question asking for various examples of modules. Besides the free modules, the next-easiest R-modules for a ring R are ideals I and quotient rings R/I. In particular, remembering that abelian groups are Z-modules is useful.
The standard "geometric" principal ideal domain is k[X], for k a field. The standard "geometric" UFD is a polynomial ring over a field (or over a UFD). So if you want a UFD that isn't a PID, you have a bunch of choices, like k[X, Y] or Z[X]. If you want an integral domain that isn't a UFD, you can think of the coordinate ring of a generic affine variety.