The following braid applet lets you play with the braid group B_5.
A sequence of letters you type in is called a "word" (in braid theory, if not in Scrabble). The program responds with a picture of a "braid". Now that we have wikipedia, there's no need for me to explain exactly what a braid is. However I will point out that two different words can represent the same braid. Some important examples are aA = 1, ac = ca, and aba = bab.
Patrick Dehornoy and his student Jean Fromentin have vastly extended the functionality of my simple little program. Djun Kim also has a java program illustrating a different viewpoint on braid groups. Good textbooks include "Braids, Links and Mapping Class Groups" by Birman, and "Knots, links, braids and 3-manifolds" by Prasolov and Sossinsky.
Roughly speaking, my job is to turn braids into matrices, which are just arrays of numbers with a complicated rule for multiplication. I am interested in finding matrices that multiply in the same way as braids do. For example, aba and bab represent the same braid in B_5, so my matrices for a and b should be such that multiplying out a times b times a gives the same answer as multiplying out b times a times b.
What's all this good for? Like most mathematicians, I do it because it is beautiful and interesting - more than this short explanation could convey. I take comfort in the fact that braid groups are sufficiently "mainstream" to be useful to plenty of other mathematicians, and hence indirectly to unforseeable future applications. Turning things into matrices is a powerful way of studying rotations of highly symmetric objects, which is useful for all sorts of things. There are also connections to quantum mechanics, which it would be nice if I understood better (or at all).