In the modern approach using scheme theory, Weil's approach is
encapsulated in the following observation. Given a finite type scheme X
over K and a point x in X, the residue field K(x) of the local ring
O\_{X,x} is a finitely generated field extension of K, hence it admits an
embedding K(x) --> Omega (sorry I seem to have a problem with encoding and cannot get Omega to work properly in this answer box). This corresponds in turn to a morphism Spec(Omega)-->X with image x. In other words, the map X(Omega) --> |X| from the Omega-points of X to the set of points of the scheme X is **surjective**.
You can then take the preimage of the Zariski topology on |X| to X(Omega) and get a topology on X(Omega). What Weil did was basically to work directly with this topology.
I would recommend learning directly the language of varieties and then schemes, and not spend any time with Weil's approach until you can make sense of the previous paragraphs - and then you will see that this approach, while historically important, is completely superseded by the language of schemes.
