I think of monads in terms of algebraic theories.

Monads are substantially more general than this intuition suggests! Here is a better intuition: monads are categorified idempotents.

The point of idempotents (acting on, say, a module) is to pick out nice subobjects: subobjects that are so nice that they are simultaneously subobjects and quotient objects (say, direct summands of modules) in a compatible way. More formally, every idempotent $m : X \to X$ wants to become a pair of a map $f : X \to Y$ and a map $g : Y \to X$ such that $g \circ f = m$ and $f \circ g = \text{id}_Y$. Similarly, the point of monads is to pick out nice categories which simultaneously map into and out of a category in a compatible way (via an adjunction). More formally, every monad $M : C \to C$ wants to become a pair of a functor $F : C \to D$ and a functor $G : D \to C$ such that $F$ and $G$ are adjoint and $G \circ F \cong M$.

This analogy is quite robust: for example, the analogue of taking the fixed points of an idempotent is taking the category of algebras of a monad. And the analogue of an adjunction being monadic is a submodule being a direct summand.

Descent on the other is supposedly of geometric nature; a formalism which generalizes familiar gluing over open subsets to a more general setting without a spatial topology. I can't imagine how or why these two notions should be related.

Here's a simple toy model. Let $f : X \to Y$ be a map of sets and let $\text{Sh}(X)$ be, for concreteness, the functor assigning a set $X$ the category of sheaves of sets on $X$, which just means the category of assignments, to each $x \in X$, of a set $A_x$. There is a pullback functor

$$f^{\ast} : \text{Sh}(Y) \to \text{Sh}(X).$$

It has a right adjoint $f_{\ast} : \text{Sh}(X) \to \text{Sh}(Y)$ given by taking fiberwise products: that is, if $A$ is a sheaf of sets on $X$, then

$$f_{\ast}(A)_y = \prod_{f(x) = y} A_x.$$

(It also has a left adjoint which we'll ignore.) This adjunction induces a comonad $f^{\ast} f_{\ast} : \text{Sh}(X) \to \text{Sh}(X)$ sending a sheaf $A$ of sets on $X$ to the sheaf

$$f^{\ast} f_{\ast}(A)_x = \prod_{f(x') = f(x)} A_{x'}.$$

Now, what should descent mean in this situation? $f$ should be descent iff it is surjective, and descent should intuitively say that a sheaf on $X$ descends to a sheaf on $Y$ iff for all $y \in Y$, all of the sets $A_x, f(x) = y$ are canonically identified. This reflects the fact that $f$ is surjective iff $Y$ itself is obtained from $X$ by quotienting by the equivalence relation $x \sim x' \Leftrightarrow f(x) = f(x')$. (Which in turn says that surjections in $\text{Set}$ are effective epimorphisms.)

This is encoded by the comonad above as follows. A coalgebra for the above comonad is a sheaf $A$ on $X$ together with a map $A \to f^{\ast} f_{\ast}(A)$ satisfying some compatibilities. What does such a map look like? Stalkwise it looks like a map

$$A_x \to \prod_{f(x') = f(x)} A_{x'}$$

and this map will end up encoding a bunch of isomorphisms $A_x \cong A_{x'}$. These isomorphisms should satisfy a cocycle condition which is encoded by the coalgebra compatibilities.

The decategorified version of this story is that a real-valued function $A : X \to \mathbb{R}$ descends to a function $Y \to \mathbb{R}$ iff $A_x = A_{x'}$ for all $x, x'$ such that $f(x) = f(x')$. Moreover, if $f$ has finite fibers, we can pick out which functions these are as the fixed points of the idempotent

$$m(A)_x = \frac{1}{|f^{-1}(f(x))|} \sum_{f(x') = f(x)} A_{x'}$$

acting on the vector space of functions $X \to \mathbb{R}$.

A more interesting case to work out is the case that $f : X \to X/G$ is a Galois cover with Galois group $G$. In this case descent will say that $\text{Sh}(X/G)$ is the category of homotopy fixed points $\text{Sh}(X)^G$ for the action of $G$ on $\text{Sh}(X)$, and the way in which the comonad $f^{\ast} f_{\ast}$ encodes this fact is a categorification of the fact that if $G$ is a finite group acting on a vector space $V$ (over a field of suitable characteristic), the subspace $V^G$ of fixed points is a direct summand picked out by the idempotent $\frac{1}{|G|} \sum_{g \in G} g$. This is a geometric form of Galois descent.

Descent in triangulated categoriesby Paul Balmer (even if you aren't interested in triangulated categories). It has a very readable introduction in which the basic setup of monadic descent theory is recalled. $\endgroup$ – Ingo Blechschmidt Mar 28 '16 at 11:39