There are loads of different types of algebras around. Start to list here for future ref.
Some introduction is in MacLane: Homology. In Homol alg folder
Skowronski ed: Trends in Representation Theory of Algebras and Related Topics, EMS
Central simple algebras: See Grillet
Jordan algebras: See Bourbaki exp 31
Chiral algebras: See Beilinson book
Hecke algebras
Vertex operator algebras
http://ncatlab.org/nlab/show/von+Neumann+algebra
Book by Chevalley on giga: The construction and study of certain important algebras
nLab page on Algebras