A kata is a series of movements in a number of martial arts (also called 'forms' and a million other names depending on the style and origin). You practice the same movements over and over focusing on technique, balance and your state of mind.

As referenced in in the story there are often a number of different types to choose from depending on what you are trying to achieve (relaxation, a specific move/technique, different fight situations). They also generally get harder and more refined as you go up belts/levels.

Never heard of it before in the coding world though, might take it up, definitely looks useful!