A topology is defined by one thing: the definition of openness. Formally, a topology is a set X (the "universe") and a collection T of subsets of X (the open sets). A simple enough definition, but what's the intuition?
The most straightforward topologies to use are the real topologies, where the space is Rn (often R2 or R3) and the open sets are the open intervals (x, y) or open solids like discs -- which have blurry edges, as opposed to circles which have hard edges. These serve as a good visualization for the basic concept that an open set doesn't contain hard boundaries; for any particular point x in the set, there is a little more "wiggle room" left between x and the edge of the set.
Topology deals with transformations that preserve certain aspects of a set; you'll often hear people use the intuition that you can stretch, twist, and bend an object, but you can't tear or glue it. Homeomorphisms are transformations that preserve the relative "nearness" of points; when you stretch something, you might make two points farther from each other, but they'll still be closer to each other than points that have been stretched even farther away.
In particular, the definition of continuity in topology is a nice generalization of the definition in calculus: a function is (topologically) continuous if the inverse image of an open set is open. Restricting this to R2, for example, it fits with the intuition from calculus that there are no "edge points" or "breaks" in the curve of the function wherever there are no breaks in the domain. In other words, a function on R2 takes open intervals on the x-axis and "stretches" them to the shape of the function's curve, but never breaks them.
I've never been very good at the geometric visualizations of topological concepts, but what's cool about it is that abstract constraints about the preservation of structure of an object lead to striking visualizations with two very different objects that yet have similarity that you can still make out. To see that a coffee mug and donut are somehow "the same" requires what the philosopher Walter Benjamin would call the human "mimetic faculty". But this really isn't surprising if you realize that abstract concepts like homeomorphism ignore many of the properties that distinguish the two objects, and only focus on those properties they both share.
(I love reading math entries in Wikipedia. Peter has pointed out glaring mistakes to me before, but I remain loyal. They tend to strike a great balance between formality and intuition, and I often find their entries more useful than Mathworld.)