At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions.
Set up: Let $k$ be a field (not necessarily algebraically closed).
Take any $f_1,\ldots,f_m \in k[X_1,\ldots,X_n]$ and define $$V=\{(a_1,\ldots,a_n)\in k^n : f_i(a_1,\ldots,a_n)=0 \text{ for } i=1,\ldots,m\}$$ Such $V$ is said to be an affine variety (in my sense).
1) What is meant by a "closed point" of $V$?
2) If $x$ is a closed point of $V$, what is meant by the "residue field" of the $V$ at $x$?
If that's your definition of a variety, every point is a closed point. In the scheme-theoretic setting, varieties also include "generic points," which correspond to higher-dimensional subvarieties, and moreover it also includes additional closed points that you don't see with your definition.
The local ring of $V$ at $x$ -- that is, the ring of rational functions on $V$ where the denominator doesn't vanish at $x$ -- is a local ring, meaning is has a unique maximal ideal (namely, those functions whose numerators vanish at $x$). The quotient by this ideal is called the "residue field." (These are more interesting when you're working the scheme-theoretic setting and can see all the extra points of the variety that you're not seeing right now.)
To make sense of this, you probably have to go ahead and really learn what a scheme is first.