A binary operation on a set$S$ is a function from $S \times S$ to $S$. A magma (binary algebraic structure) is a set equipped with a binary operation on it. A magma is unital if it has a neutral element$1$. Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1). The Eckmann-Hilton argument holds for unital magma structures: two compatible ones on a set must be equal and commutative.

More generally, in any multicategory$M$, a magma object or magma in $M$ is an object$X$ of $M$ equipped with a multimorphism$m: X, X \to X$ in $M$. Here the multimorphism from $X$ and $X$ to $X$ is a binary operation in $M$. In particular, for $M$ a monoidal category, a magma structure on $X$ is a morphism$m\colon X \otimes X \to X$; and in a closed category, a magma structure on $X$ is a morphisms $m\colon X \to [X, X]$.

Literature

The wikipedia entry magma is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.