So, the definition of is that .
Whew, that's a mouthfull, so let's break that down some. Firstly, the condition is say that for any positive number , (no matter how super-small!), I can come up with another number so that when my x is a -distance away from c, will be within -distance from L.
Think of the condition like a game. Someone gives you a number . All you know about it is that it's positive. You get no other information. What you're trying to do is pick a (you pick ) so that the condition of the limit is satisfied.