Thompson is most famous for his work on the Feit-Thompson theorem, that every group of odd order is solvable. Solvable groups resemble upper triangular matrices: a solvable group is constructed in layers out of abelian groups.
Tits invented the notion of BN pairs and buildings. The opposite of a solvable group is a simple group, which cannot be split up into layers. Simple groups tend to resemble the set of all invertible n-by-n matrices over a field (which itself is not simple, but is pretty close to it). Tits identified the key property that makes the resemblence work: the existence of special subgroups B and N. For the group of invertible matrices, B is the set of upper-triangular matrices, while N is the set of permutation matrices. Buildings are a geometric explanation of BN pairs.