Let $L/K$ be a Galois field extension and consider a variety $Y$ over $L$. The theory of (Galois) descent addresses the question whether $Y$ can be defined over $K$. More precisely, the question is: "does there exist a variety $X$ over $K$ such that $Y = X \times_{Spec(K)} Spec(L)$".
Now assume such $X$ does exist. In this case $Y$ is endowed with $Gal(L/K)$ action coming from an action on the second factor.
Conversely, if $Y$ has a Galois action compatible with the action on $Spec(L)$, then $Y$ descends to some $X$ defined over $K$. $X$ is actually a quotient of $Y$ by $Gal(L/K)$ (so that the conjugate points glue together to form one point on $X$). Note that the set of $K$-points of $X$ is the set of Galois fixed points.
Example. $K = \mathbf R$ and $L = \mathbf C$. For any real variety, the set of complex points admits the action of $ mathbf Z/2$ by complex conjugation. Conversely, if a complex variety is endowed with conjugation, it descends to a real variety.
Remarks. Theory of descent also classifies all possible $X$'s arising from $Y$. Such $X$'s are called forms of $Y$. They are in 1-1 correspondence with a certain Galois cohomology group.