Basically, a limit is what it sounds like. A function f(x) has a limit, say b, at x = a if f(x) become closer and closer to b as x becomes closer to a.
The value of f(x) at x = a is irrelevant to whether f has a limit at a. However, if f does have a limit b, then putting f(a) = b makes f continuous at a. So, if f cannot be made continuous at a, it does not have a limit.
Another way to visualize this is to say that you can zoom indefinitely at x = a and still fit the graph of f in some neighborhood of x = a inside your screen. If, at some point, the graph goes through the top or bottom of the screen regardless of how small you make the segment around x = a, the function does not have a limit at a.