A code kata is a simple coding problem intended to be easily solved, that you solve again and again, repeated to perfection. You learn to solve the problem better every time you try, and you will also learn to write code faster, using more shortcuts and hotkeys if you try. Another benefit is that your subconscious will learn detailed problem/solution pairs that will help you in your everyday coding. You can also try to solve a kata using different programming languages or with limitations such as "without using the mouse" to further challenge yourself.
Here I will publish links to Code Katas around the web. If you know of any Kata that isn't linked already, please add a comment. Please add a comment also if there are any specific kata(s) you really like.