What does the cardinality of a vector space depict? I always assumed that the concept of cardinality is only applicable to sets. Yesterday I read somewhere that the cardinality of a vector space $V$ over a finite field $F$ with $q$ elements equals $q^{\dim V}$. 
What exactly does this number represent?
 A: Mathematical objects are$^1$ sets with some additional structure. E.g. a group is a set with a binary operation such that [stuff], a ring is a set with two binary operations such that [stuff], and a vector space over a field $k$ is a set with a binary operation, and a unary operation for every $x\in k$, such that [stuff].
When we speak of the cardinality of a mathematical structure, we're referring to the cardinality of the "underlying set" (which is variously called the "carrier set", "domain", "universe", and probably a bunch of other things).
As to the question of how we calculate the cardinality of a vector space over $k$ - well, we're counting the vectors! So let's say I have a field $k$ of cardinality $\kappa$. Then if my vector space $V$ is $n$-dimensional, I can represent a vector in $V$ as an "$n$-tuple" $(a_1, . . . , a_n)$ of elements of my field! So in particular, the number of vectors is $\kappa^n$.
It turns out we can make sense of this even if the field is infinite, or the space is infinite-dimensional; but that starts getting into some mild set theory.

$^1$Hoo boy. This'll get me in trouble.
The fact is that the statement "Mathematical objects are sets" is a hugely loaded proposition, one that many reasonable mathematicians disagree with (including me, depending when you ask). I don't want to get into that here, since that's a topic for another question; but I do want to point out that there is a lot more to that claim than meets the eye. If you're interested, here's a few terms you might want to look up:


*

*Set-theoretic foundations

*Structural set theory

*Non-concrete categories
But these are fairly advanced topics, and might best wait 'till later. Tl;dr, it's not really fair to just say "mathematical objects are sets," but right now it's probably a decent approximation of the truth.
