Probably the simplest way to understand the concept of a continuous function is to think of real functions of a real variable. Something like x2
Here, the word continuous means that the graph of a function should have no breaks. Note that continuity is not a smoothness property. |x| is a continuos function but it has a wedge at the origin. The step function is discontinuous. 1/x is discontinuous at zero, because on one side of zero it goes off to +infinity and on the other side it goes of to -infinity.
Now lets consider a function of two variables. Here the function would trace out a surface. Again, we say that it is continuous if the surface has no breaks in it.
All this thing about epsilons and deltas essentially means that if the function has one value at a point, and then if I move slightly away from that point, the value of the function should not change drastically!