ag.algebraic geometry - What is etale descent?

Let $K/k$ be a finite separable extension (not necessarily galois) and $Y$ a quasi-projective variety over $K$.

The functor $k-Alg to Sets:A mapsto Y(Aotimes_k K)$ is representable by a quasi-projective $k$-scheme $Y_0=R_{K/k}(Y)$.
We have a functorial adjunction isomorphism
$Hom_{k-schemes}(X,R_{K/k}(Y))=Hom_{K-schemes}(Xotimes _k K,Y)$

and the $k$-scheme $Y_0=R_{K/k}(Y)$
is said to be obtained from the $K$-scheme $Y$ by Weil descent.
For example if you quite modestly take $X=Spec(k)$, you get
$(R_{K/k}(Y))(k)=Y_0(k)=Y(K)$, a formula that Buzzard quite rightfully mentions.
If $Y=G$ is an algebraic group over $K$, its Weil restriction $R_{K/k}(G)$ will be an algebraic group over $k$.

As the name says this is due (in a different language) to André Weil: The field of definition of a variety. Amer. J. Math. 78 (1956), 509–524.

Chapter 16 of Milne's online Algebraic Geometry book is a masterful exposition of descent theory, which will give you many properties of $(R_{K/k}(Y))(k)$ (with proofs), and the only reasonable thing for me to do is stop here and refer you to his wondeful notes.