As noted by Zhen Lin, monads (on $\mathbf{Set}$) should be thought of as morally equivalent to *algebraic* theories of fairly general type. By an algebraic theory I mean a single-sorted theory specified by operations and universally quantified equations between terms. The operations are allowed to be infinitary. Certainly all monads on $\mathbf{Set}$ and their algebras are describable by algebraic theories and their models. Not quite all algebraic theories are thus describable by monads, but all algebraic theories whose signature consists of a small set of operation symbols are "monadic", and there are some notable examples of monads where the signature of any algebraic theory giving rise to it must consist of a proper class of operation symbols.

First, here are some non-examples. The theory of fields is *not* algebraic in this sense, since the multiplicative inversion is not a globally defined operation. (It's not algebraic even if we allow multi-sorted theories. For the category of fields does not, for instance, have products -- whereas this would be true of any category of models of a multi-sorted theory.) Nor is the theory of categories algebraic in this sense -- while it can be presented as a single-sorted theory (where the sort gets interpreted as the set of morphisms), the composition is not given by a globally defined operation. (And again, it's not even algebraic in any multi-sorted sense -- if it were, then the category of models would be a regular category, which $\mathbf{Cat}$ is not).

To get an algebraic theory out of a monad $M$ on $\mathbf{Set}$, one may take the Kleisli category $Kl(M)$, viz. the full subcategory of the category of $M$-algebras whose objects are free algebras. Then define an infinitary theory $Th(M)$ by taking operation symbols of type $n \to 1$, for $n$ a possibly infinite cardinal, to be elements of the (underlying set of the) free algebra $M(n)$. This theory should be thought of as "saturated" (aka a "clone"), consisting of all definable operations of a theory; the point is that the monad multiplication $\mu: MM \to M$ takes a term $\alpha \in MM(n)$ (obtained by formally substituting $k$ operations $n_1 \to 1, \ldots, n_k \to 1$ into some $k$-ary operation, to arrive at a term of arity $n = n_1 + \ldots + n_k$) and returns an operation symbol $\mu(\alpha) \in M(n)$, which is equated to that definable term $\alpha$ in the theory.

If you like thinking in terms of Lawvere theories, then you could say that $Kl(M)^{op}$ plays the role of the infinitary Lawvere theory attached to $M$: the category of $M$-algebras is equivalent to the category of functors $Kl(M)^{op} \to \mathbf{Set}$ that preserve arbitrary products.

If the monad $M$ is *finitary* (preserves filtered colimits), then we can cut down to the usual Lawvere theory given by free algebras on *finite* sets. This generalizes: if $M$ preserves $k$-filtered colimits for some regular cardinal $k$, then one can cut down to a generalized Lawvere theory $Th_k(M)$ given by free algebras on sets of cardinality less than $k$. Then the category of $M$-algebras is equivalent to the category of functors $Th_k(M) \to \mathbf{Set}$ preserving $k$-small products. Such monads correspond to theories of rank $k$.

Not all monads $M$ on $\mathbf{Set}$ are $k$-accessible in this sense; a famous example is the category of compact Hausdorff spaces, whose underlying functor to $\mathbf{Set}$ is monadic, but where the corresponding algebraic theory is too large to have rank.

Quite a lot of this material is worked out in the nLab.