They look at this elliptical plate on the table, and think it's a circle, because they know what happens when you look at things at an angle like that. In topology, any continuous change which can be continuously undone is allowed. These two process are continuous in the sense that during each of them, nearby points at the start are still nearby at the end. At least, this is true if you use clay which is still soft and hasn't been fired yet. Once they're fired they become Euclidean rather than topological, because you can't flatten the bowl any longer without breaking it.
Topology is almost the most basic form of geometry there is.
It is used in nearly all branches of mathematics in one form or another. There is an even more basic form of geometry called homotopy theory, which is what I actually study most of the time. We use topology to describe homotopy, but in homotopy theory we allow so many different transformations that the result is more like algebra than like topology. For a more extensive essay on this topic with charming diagrams, see Neil Strickland's answer, and for an index of related essays, see Don Davis's list. 2000 Robert R. Bruner.