Number of solutions to congruences

Is there any general form to determine the number of non-congruent solutions to equations of the form $f(x) \equiv b \pmod m$?

I solved a few linear congruence equations ($ax \equiv b \pmod m$) and I know those have only one solution because we're basically finding $a^{-1}$ and all the inverses of $a$ are congruent.

What's the number of solutions for congruences of higher degree polynomials? (quadratic, qube, etc).

Thanks a lot.