What are some examples of "exotic" algebraic structures?

Question

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.

Answer 1

Quandles arise (allegedly) quite naturally in knot theory. They are also connected to group theory, since the conjugation operation on a group gives rise to a quandle.

Answer 2

You might be interested in Universal Algebra. You could build your own algebra that way. Have a look through A Course in Universal Algebra, by S. Burris and H. P. Sankappanavar; it builds up a wonderful theory for them. Examples from this book include "squags" and "sloops".

Answer 3

Groupoids seem somewhat seem somewhat more exotic than groups, but are actually a lot more natural in many ways. For example the fundamental groupoid is really more "fundamental" than the fundamental group in many ways.

Operads are also quite handy when doing algebraic topology. I recommend looking at Tom Leinster's book "Higher operads, higher categories".