TITLE: Does cardinality really have something to do with the number of elements in a infinite set?
QUESTION [5 upvotes]: I've seen some videos and read some texts (non-rigorous ones) that explained the concept of cardinality, and sometimes I see someone asking if there are more numbers between the reals in $[0,1]$ then numbers in the set of natural numbers. I've read about uncountability in several places and it seems that:

The interpretation that there are more numbers is vague and inaccurate;
That the real point is about the diagonalization of the members of a set, if such a task is possible or no;
And that the idea of having more numbers is a kind of simplification that they give to the laymen;

So does cardinality really have connections with the notion of quantity of elements of a set or not? I know (I guess) that the cardinality of a finite set is the counting of the number of elements on it and perhaps this notion has leaked somehow to the notion of cardinality on infinite sets which seems to be a very different idea.
I feel that the idea of the quantity of elements of an infinite set is weird per se, when I read about the diagonalization, it made a lot more sense than the idea of quantity of an infinite set.

REPLY [5 votes]: The set-theoretic perspective has been well represented in other answers, but there is another perspective that has not been mentioned yet. 
Although it is not emphasized in set theory contexts, the use of bijections to measure cardinality is closely related to "Hume's principle", which is the claim that if two collections $A$ and $B$ of objects are equinumerous, then there is a bijection between them. In this setting, "equinumerous" means that there is a relation $R$ such that every $a \in A$ is related to exactly one bijection in $B$, and every $b \in B$ is related to exactly one object of $A$. For a careful treatment of this background, see Frege's Theorem for Foundations of Arithmetic by Edward N. Zalta in the Stanford Encyclopedia of Philosophy.
The distinction between cardinality and equinumerousity may seem to be only a trivial distinction, but that appearance is only because we are used to working in set theory with the full axiom of separation and the axiom of choice. When we work in other settings, Hume's principle becomes much less straightforward. 
It could equally well be argued that the conception of cardinality taught in ordinary set theory is the relational one (the one that, by Hume's principle, is equivalent to the bijectional definition of cardinality). Because these are so tightly intertwined in ZFC, where relations are reduced to just another kind of set, it is hard to learn one of these concepts of size without learning the other.  
One way that Hume's principle can fail is when we take "relations" to mean "definable relations" and we limit the set existence (comprehension/replacement) scheme. This happens in settings including Kripke-Platek set theory and systems of higher-order arithmetic. In those settings, there is no reason to think that just because a formula $\phi(x,y)$ defines a one-to-one correspondence between two sets in a model, that there must actually be a function in the model that witnesses the correspondence.