To flinch is to react to pain by wincing, pulling away, or cringing. Often it's used to describe a person who shows a moment of weakness or fright: he was so tough, I thought he'd never flinch, but snakes really freak him out.
To flinch is to pull away suddenly or recoil when something frightens or hurts you. People flinch at different things. Needles, for example, cause some people to flinch, while they're not even noticed by others. If you can withstand something awful without showing it, you're likely to hear someone say "Wow! You didn't even flinch!" Think of it as a combination of flee and pinch. If you don't flee when you get pinched, you don't flinch.