What does a topologist do?
Topology is the study of geometric shapes, with regard only to those properties that are unchanged by stretching and bending. Geometry deals with properties such as distances and angles, which are of course changed if you stretch or bend the objects. In topology we ignore these kind of properties. From the viewpoint of topology, a square is the same as a circle, since you can deform one into the other! On the other hand, if you have a circle in space, you cannot turn it into a trefoil knot without tearing it. Therefore, the circle and the trefoil are topologically different:
One part of topology is knot theory, which classifies knots (closed loops in space). The Knot Atlas is a comprehensive resource about this subject. There is also a YouTube video introducing knot theory.
Although knots live in three-dimensional space, we topologists call them one-dimensional objects, because an ant living on the knot has only one dimension in which to go: along the knot. Similarly, we say that the surface of a sphere is a two-dimensional object, because an ant living on the sphere can go around in two directions.
An amazing topological property of the sphere is that it can be turned inside out, as shown in this video.Here are a few more two-dimensional shapes, all topologically distinct from the sphere:
These are called the torus, the triple torus, the Möbius band, and the Klein bottle. The last two are interesting because they have only one face. You can order a Klein bottle made of glass at ACME Klein bottles. I have one in my office.
Geometric shapes get even more interesting (and complicated) when you go into higher dimensions. A particularly active area of current research is the study of four dimensional shapes.
However, it doesn't matter. We topologists study abstract shapes, without regard as to whether they appear in the physical universe. These shapes are still useful as mathematical models for various purposes. For example, if an economist wants to describe the economy in terms of the prices of 28 commodities at any given time, the prices can be viewed as a point in a 28-dimensional space.
It is quite difficult to imagine a 28-dimensional shape. The power of topology is that it gives methods to study such shapes, even if we can't visualize them directly.
This theorem is remarkable because it was the first major result in mathematics that was proved (in 1976) using a computer. The proof is too complex and laborious to be understood by any human without the use of technology.
Another famous problem in topology is the Poincaré conjecture. It says that the three-dimensional sphere is the only smooth, closed, simply connected, three-dimensional shape. (Closed means that the shape is finite and does not have boundary. Simply connected means that every loop in your space can be contracted into a point. In other words, if you throw a lasso onto your shape, you can't really catch anything.)
The Poincaré conjecture took 100 years to be solved. It was finally proved in 2002 by the reclusive Russian mathematician Grigori Perelman. There was a million dollar prize for a solution, but Perelman refused it. You can read more about the story around the conjecture in Donal O'Shea's book.
Four-dimensional shapes are more difficult to understand than shapes of any other dimension. Of course, it makes sense that they are more complicated than two or three dimensional objects, but the interesting thing is that they are also more complicated than 47-dimensional objects. Roughly, the reason is that in higher dimensions there is more room to move around, so one can pull apart the parts of our objects that are complicated, and study them separately.
Another interesting problem is to be able to tell if a given knot can be unknotted. Actually, this problem was solved by Haken in 1961, but finding better algorithms is still an active area of research. I worked on this problem myself (with my collaborators Peter Ozsváth and Sucharit Sarkar) and we found a new algorithm in 2006. Here are some slides from a talk I gave describing this work.
To be honest, many of these applications use only "classical" topology, that is, the kind of topology developed 50 or even 100 years ago. Most of the recent results in topology do not yet have any practical applications. However, this is how basic research goes: it comes way before it finds any use in the real world.
There are also those of the opinion that math is more an art than a science, and that discovering new mathematical structures has an intrinsic, aesthetic value. G.H. Hardy, who worked in number theory (another abstract branch of math) in the first half of the twentieth century, was proud to say: "Nothing I have ever done is of the slightest practical use." Nevertheless, nowadays number theory is of utmost importance in cryptography; it is the foundation of internet security. It is quite possible that the topological theories we develop today (such as Floer homology) will never find practical applications, but it is also very possible that they will.
If you are interested in learning topology more seriously, here are some of the standard textbooks. They are arranged in the order of difficulty, from Weeks' book (aimed at high school students) to Hatcher's (aimed at graduate students):