Is it true that two distinct irreducible algebraic curves intersect only at points?
I have two irreducible bivariate polynomials and would like to say something about the geometry of their shared zeros.
Sure, this is a consequence of Bezout's theorem which also tells you how many points there are if you count them carefully enough to account for complex roots, multiplicities and points at infinity. The theorem requires that the curves don't share a component, but this is guaranteed by their being irreducible (and distinct).
Bezout's theorem is about smooth projective connected curves.
It might happen that there aren't any solutions. Take for example f(x) = x-1 and f(x) = x.
Maybe I'm not understanding the question though.