Let $f: U \rightarrow X$ be a faithfully flat morphism of nice schemes (quasiseparated, quasicompact, and anything else I might have forgotten). One can understand descent in quasicoherent sheaves using Barr-Beck for comonads. The conditions on the theorem are a mild condition on existence of certain equalizers (not an issue) and that $p^*$ is conservative, which (I think) is satisfied by faithfully flatness of $f$. So we can apply Barr-Beck.
Barr-Beck says that $QCoh(X)$ is equivalent to coalgebras over the comonad $T = f^* f_*$ in $QCoh(U)$. By flat base change, the monad $T = (p_2)_*(p_1)^*$ where $p_i$ are the projections $U \times_X U \rightarrow U$. A comodule is the data of an object $M$ and a morphism $M \rightarrow T(M)$. Using the adjunction this morphism is the same as a morphism $\sigma: p_2^* M \rightarrow p_1^*M$.
I'm having some trouble unwinding the coherence relations for coalgebras to get them to agree with the usual cocycle condition for descent. Here is what I want the coherence relations to say. There are three pullbacks of $M$ to $U \times_X U \times_X U$ and also three ways to pull back $\sigma$ to the category of qcoh sheaves on $U \times_X U \times_X U$. The easiest way for me to see this is using a 2-simplex:
So, on each 0-vertex we have one of the pullbacks of $M$, on each 1-edge we have one of the pullbacks of $\sigma$, and in the middle we have strict equality, i.e. these maps are required to commute on the nose. This should be the cocycle condition (i.e. that they commute on the nose).
Just for reference, the coherence relations for comodules can be expressed by the commuting of the split equalizer:
Aside: In particular, the cocycle condition implies that $\sigma$ is an isomorphism; if we think about what this means when restricting to the (1,2,3)-diagonal (points that look like $\{(a,a,a) \mid a \in X\}$, or "triple intersections of the same open"), all the pullbacks agree, so $\sigma$ is the identity. If we think about the $(1,3)$-diagonal (i.e. triple intersections where the first and last open are the same), the bottom edge has to be the identity, and the other two edges are just $\sigma$ on the double intersections and this say that the morphism is invertible, i.e. isomorphism.