The universal algebra perspective on monads is that m x is the set of expressions (ie. ASTs) for a particular algebraic structure with variables drawn from the set x. For example, you have a monad for abelian groups where the expressions are things like a+b, 2a-b, 1. Note that m encodes both the operations of the algebraic structure, like +, and the laws that they obey, because we regard a+b and b+a to be the same AST (ie. these are considered equal as a matter of syntax).

A concrete algebraic structure is a map m x -> x that evaluates expressions, producing their value, eg. 7+3 -> 10. It essentially provides concrete definitions for the syntactic operations in the ASTs.

Under this interpretation, the monad operations say "a bare value is an expression, eg. a, b" (x -> m x) and "nested expressions can be simplified, eg. (a+b)+(c) = a+b+c") (m m x -> m x). The monad laws essentially are ensuring that no matter what order you evaluate subexpressions and simply an expression, it always comes out to the same value in the end. It essentially lets us state a Church–Rosser theorem for the maps m x -> x.