A Kleinian group is a type of mathematical group that was first studied by a mathematician called Felix Klein. Groups are collections of things that have special properties. With a Kleinian group, the things that are in the group must obey certain rules.
One of these rules says that if you pick two things from the group and combine them, then you must get something that is back in the group. So if you pick two things from the group, combine them together, then you should get back one of the things you started with.
Kleinian groups are special because no matter how many times you take two things from the group and combine them together, at the end you will always get the same thing back.