My approximate understanding: Some problems reduce to calculating the trace of a matrix. The matrix might be really complicated, but sometimes you can approximate it by having particles braid around each other and measuring how likely the particles end up in a particular state. In theory it can be put into practice using materials called "topological insulators."

The Jones polynomial is a thing which can tell knots apart. The quantum computer can calculate the Jones polynomial by having particles trace out the knot. The Jones polynomial usually takes exponential time to calculate.

The "topological" in "topological quantum computing" refers to the braiding and the fact that they're computing topological invariants of a space.