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The concept of limit says that as x approach c the function f(x) tends to L. That is the less the distance from c the closest the function to L. Now, think of epsilon and delta as "error" and "distance" respectively. Epsilon is the desired level of proximity of the function to L and delta the distance from the point c (on both side of the real line in one variable, hence the absolute value). The delta -epsilon definition can be thought as a game. The opponent pick an error (epsilon) and I will always be able to find a distance for which all the points within that distance will have a f(x) value closer to L. It doesn't matter how small the error, I will always find a suitable distance, because the closest x is to c the nearest the function is to L. Hope it doesn't make more confusion.
In same line as zoolgoth, Please read Conversation given Below.. AamirKhanday: Hey Raghav, I(f) am very sad. Now, My(f) desired degree of closeness with L is epsilon. Can you help me in finding any more x's ? Raghvendrac: Hello Aamir, Don't worry friend. Don't you remember.. "L is your Limit at c". I can always give you infinitely many points for any of your desired closeness with L. I can always find infinitely Many close friends(Delta Nbd) of c which will work for you (To get closeness with L).
first understand what are epsilon,delta, x and f(x) then understand the concept of closeness. also the concept of infinitesimals. after understanding these concepts precisely u will be able to understand the epsilon-delta definition of limits search wikipedia and this http://archives.math.utk.edu/visual.calculus hope it will help!