Roughly, "L is the limit of f(n) as n goes to infinity" means "when n gets big, f(n) gets close to L." So, for example, the limit of 1/n is 0. The limit of sin(n) is undefined because sin(n) continues to oscillate as x goes to infinity, it never approaches any single value.
Technically, L is the limit of f(n) as n goes to infinity if and only if for every e>0, there is a b>0 such that |f(n)-L|<e whenever n>b.
Let's apply this to our first example above (the limit of 1/n is 0): Suppose e is any positive number. If we let b=1/e, then n>b is 1/n<e. This is enough to prove the limit of 1/n is 0. Look in the front of almost any Calculus textbook to see more examples of this definition, as well as the numerous other forms of limits.