Descent algebras are associative but not semi-simple algebras associated with Coxeter groups. The easiest one to describe shows what their fascination is. We partition the permutations of Sn into 2n-1 subsets according to the up-down shape of each permutation (e.g. 3 4 5 1 2 has shape 'up','up', 'down', 'up'). Now add up the elements of each class (working in the group algebra). Miraculously, you get the basis of an associative algebra with integer structure coefficients!
The study of these algebras began in the 1970s and is still very active. My part in it was some contributions towards the representation theory in characteristic zero, and in characteristic p.