What is your favorite non-UFD domain and why?

My favorite one: If R is the localization of k[x, y, z]/(x^2+y^3+z^7) at the prime ideal (x, y, z), then the ring of formal power series R[[x]] over R is not a UFD.

Another nice example is the ring of holomorphic functions in a single complex variable. This is not a UFD since there exist holomorphic functions with an infinity of zeros, and thus an infinity of irreducible factors. (In a UFD factorization must be finite.)

The most famous example however is Z[-5] since 6 = (2)(3) = (1 - i sqrt(5))(1 + i sqrt(5)) and none of the factors are associates, as you can easily verify.