First consider the case of a function f that maps a real-valued variable x to a real-valued variable y. A function y = f(x) is continuous at x=x0 if the following holds: For any positive number epsilon, there exists some positive number delta such that for all x where x0-delta<x<x0+delta, the value of f(x) will satisfy f(x0)-epsilon<f(x)<f(x0)+epsilon. More intuitively, we can say that if we want to get all the f(x) values to stay in some teeny neighborhood around f(x0), we simply need to choose a small enough neighborhood for the x values around x0, and we can do that no matter how teeny the f(x) neighborhood is.
An example of a discontinuous function is f(x)=1 if x>0, f(x)=0 if x<=0. There is no delta-neighborhood around x=0 that will force all the f(x) values to be within epsilon of f(0), for any epsilon < 1. Intuitively we can think of a discontinuity as a sudden jump in function values.
To generalize, consider a function f that maps x to y, where x is a member of a metric space X and y is a member of a metric space Y. Then f is continuous at x=x0 if for any positive real number epsilon, there exists a positive real number delta such that all x in X satisfying dx(x,x0) < delta will also satisfy dy(y,y0) < epsilon, where dx and dy are metric functions for X and Y respectively.
More generally still, we can define continuity for functions between topological spaces. Suppose f is a function from a topological space X into a topological space Y. Then f is said to be continuous at a point x in X if for every neighbourhood V of f(x) there is a neighbourhood U of x such that f(U) is a subset of V. If f is continuous at every point of X, then it is simply said to be continuous. It turns out that a function is continuous if and only if the pre-image of every open set is open, and so this is often used as the definition of continuity.
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