The intuitive meaning of "a sequence a_n converges to a limit L" is that the farther out in the sequence you look, the closer the values get to L, and more specifically,if we look far enough out in the sequence, the values get as close as we want to L (and stay at least that close forever after).

The integer N in the definition represents how far out we're looking in the sequence--that's why it says "when n>N". What the definition is really saying is: "tell me how close you want the values of the sequence to be to L, and I'll give you a number N, so thateveryvalue, after the Nth value, is at least that close to L".