>>25
Although I gave you the definition of convergence of a sequence, I guess you want the more intuitive explanation of what convergence means. You beautiful layman, you.
Essentially, as n->inf the amount that x_n may differ from x approaches 0. You could think of this as the diameter of the set of points x_n from some n going to infinity monotonically decreasing as n increases. If the diameter of a set is 0, then it is a set of one element (in this case, the value the sequence converges to).