If I recall, a commutative monoid is a monoid object in the category of monoids.

A monad is a monoid object in the category of endofunctors for some category \$$\mathcal{C}\$$.

And most impressively, a monoidal category is a monoid object in
(\$$\mathbf{Set}\$$-category)-category.

Edit: I could also be wrong and I'm getting waaaaaaaay ahead of myself.