What truly happens if you don't pay interest fees?

What actually happens to you if you don't pay the interest fees that are tacked on to your credit card and other loans? Do the banks continue to harass you or does it just cost you your credit score to plummet? I'm curious because I'm currently paying off a Care Credit bill, and despite the fact that I do work in a bank I work in processing and I'm still learning about financial topics since public schools don't do a good job teaching students about the truth.