I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, but they seem to be way to general notions as to admit a really interesting theory.
Thus, I'm wondering whether there are also other interesting algebraic structure (here, this means mainly some set $S$ together with a bunch of functions $f_i:S^n\to S$ satisfying some laws) which behave somewhat differently, i. e. satisfy some unusual relations like $(ab)c=(ca)(cb)$ or $ba=(aa)(bb)$, but in such a way that there is a decent amount of theory about them (some kind of nontrivial classification or representation theorem would be truly fascinating).
Bonus points if these structures arise naturally in some areas of mathematics.