Micromouse is a competition where you have to build a robot that can find its way through a maze. The robot that gets through the maze quickest wins. There is a website for micromouse, the website has a very good links page. The competition used to be organised by IEE but is now organised by Royal Holloway College. There is also a schools micromouse competition. There is some good information on building a micromouse at S. Benkovic's page
There is a line following micromouse competition for schools; you can see the details of this at Micromouse Wales website.
Robot Sumo is a competition in which two robots compete by trying to push the other robot out of a ring. For information on official Robot Sumo look at the pages of All Japan Robot-Sumo Tournament. There are a few different standards for this competition using different sized robots. Robot mini-sumo uses robots that are have maximum dimensions of 10c x 10cm x any height with a maximum mass of 500g. To get a view of the various options look at the home of the Northwest Robot Sumo Tournament from the USA.
Techno games involves many different competitions and is now being shown on the BBC. There is a Techno games website.