The teams are Team Boke and Team Tsukkomi, standing for the two roles in a manzai duo. In a manzai performance, the boke (funny man) will tell a story of some sort, then diverge into something nonsensical, upon which the tsukkomi (straight man) will hit the boke with something (usually a paper fan) while yelling at him or her, either about the story's sudden absurdity, about correcting the boke, or to get back on topic (or, uncommonly, something else).
Manzai's been around for hundreds of years, so there's been many different types and has increased in complexity and subjects since. It was also originally part of a New Year's Day celebration but is now a year-round thing, with stages all around Japan designed specifically for manzai. What I described above is the core of all manzai performances though.