I first learnt of magmas on Wikipedia and have been trying to read more on them just out of my own interest. Whenever I try to search them on Google, though, the search results are overwhelmed by the algebra computational system of the same name. I haven't been able to learn any more about them as a result.
If anyone has any links, keywords/search recommendations, or any book recommendations, I would really appreciate it!
Thanks!
This is probably more of an extended comment than an actual answer, but hopefully you find it useful.
Another synonym for a magma is a binar. See here: http://math.chapman.edu/~jipsen/structures/doku.php/binars
An important theorem on binars is Murskii's theorem. It (roughly) states that if you take an arbitrary finite binar, the probability of it being idemprimal is 1. Being idemprimal is property that guarantees a lot of structure on the algebra (I won't get into the details of what I mean by that). Examples of idemprimal algebras include the two-element Boolean algebra and the three-element field. It is not very intuitive that most binars have this property.
An example of interesting binars that fall outside the commonly studied semigroups, monoids, and groups are graph algebras. Given a digraph $(V,E)$ we can define a binary operation on $V\cup\{0\}$ by $xy=x$ if there is a directed edge from $x$ to $y$ and $xy=0$ if there is no edge. These binars can have many unusual properties.