Over time, the concept of Katas grew from a problem to solve to a solution to practice. Uncle Bob Martin (among others) began talking about the idea of practicing the solution to a kata until the steps and keystrokes became like second nature, and you could do them without thinking.
In TDD Katas the focus in on doing the TDD cycle, Red-Green-Refactor, right. Corey had recently published a screencast where yours truly solves the StringCalculator TDD Kata in Groovy. It illustrates some points about TDDing (like "Fake it till you make it" or "triangulation") in a very clear way.
So, If you feel like you want to better understand TDD you can watch the Kata on Corey's site. An explanation of the different moves is given in the accompanying text.
Another recommendation is to watch Gary Bernhardt's screencast where he refactors cyclomatic complexity code in Python.