Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$.
The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ which can be defined in two ways:
Definition 1.
For every $K$-algebra $A$ the $K$-scheme $W_{L/K}X$ is the representing object of the functor $$ A \longmapsto X(A \otimes_K L) = \operatorname{Mor}_L( \operatorname{Spec}(A \otimes_K L),X) . $$
Construction 2.
Taken from https://mathoverflow.net/questions/7715/what-is-restriction-of-scalars-for-a-torus .
Now for a more concrete description. Suppose $X=\operatorname{Spec}L[y1,...,yn]/J$ is an affine scheme. Let $d=[L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution": $$ y_i= a_1 x_{i,1}+...+a_d x_{i,d}, $$ thus replacing each $y_i$ by a linear expression in d new variables $x_{i,j}$. Moreover, suppose $J=\langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{i,j}]$, and we put $$W_{L/K}X=\operatorname{Spec}K[x_{i,j}]/I \ . $$
The question:
While Definition 1 is a very abstract and hard to work with object (abstract nonsense) the second definition is very practical and can be worked with. In particular, for $\mathbb{C}/\mathbb{R}$ the Weil restriction (via the 2nd construction) allows us to see an $n$-dimensional complex variety as a $2n$-dimensional real variety.
The question is how to derive Construction 2 from Definition 1.