From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used.
Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition.
Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath.
Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.
The main types of structures with one set having two binary operations are ring-like or ringoids and lattice-like or simply lattices.
Lattice-like structures have two binary operations called meet and join, connected by the absorption law.