So a circle is the same as a triangle or a square, because you just 'pull on' parts of the circle to make corners and then straighten the sides, to change a circle into a square. Then you just 'smooth it out' to turn it back into a circle. 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. The ellipse and circle are projectively equivalent.