In algebra, monads are used to reify the notion of an "explicit expression". So you have the integers, Z, and the monad m gives m(Z), the set of expressions (let's say sums) over Z. And then you have m(m(Z)) which is nested expressions and so on. An algebra for a monad evaluates an expression m(Z)->Z, and all the monad laws basically enforce the kind of equational reasoning ordinary notation takes for granted (eg sub-expressions can be replaced by their value, 1+(1+1)=1+2).

So those would be related in this way

                       join
  (1)+(1+1) : m(m(Z)) -----> 1+1+1 : m(Z)
             |                 |
  fmap(eval) |                 | eval
             v                 v
       1+2 : m(Z) -----------> 3 : Z
                    eval