A nontechnical way to comprehend $\aleph_2$ This is possibly a dumb question, but I do not know where to look for an answer.
Without getting technical, one can show why $card(\mathbb{N}) = card(\mathbb{Q}).$ (Typically by showing how the two sets are countable and by showing a one-to-one correspondence between them. In the same manner, we see why the set of real numbers is uncountable and why there cannot be a one-to-one correspondence between the set of natural numbers and the set of real numbers. Again, without getting technical, say, without the concept of power set and all that, is it possible to show why $\aleph_1 < \aleph_2?$ Why there cannot be a one-to-one correspondence between them?
Another potentially dumb question: If the set of natural numbers is countable, and the set of real numbers is uncountable, what feature does the set of $2^{\aleph_1}$ possess that the set of real numbers do not? Is that even a meaningful question?
 A: Let me tackle your second question first, since it's easier. The following is consistent with ZFC (that is, the usual axioms of set theory can neither prove nor disprove the following): $$2^{\aleph_1}=2^{\aleph_0}.$$ So even though $\aleph_1>\aleph_0$ by definition (see below for more on this), it's possible their power sets have the same size!
Re: your first question, let's begin by recalling the definition of the $\aleph$s:

An initial ordinal is an ordinal which is not in bijection with any smaller ordinal. For an ordinal $\alpha$, $\omega_\alpha$ is the unique ordinal $\theta$ such that $\{\beta<\theta: \beta\mbox{ is an initial ordinal}\}$ has order-type $\alpha$. 

Initial ordinals are also called cardinals - that is, $\aleph_\alpha$ is just another notation for $\omega_\alpha$. (This is a bit sloppy - I'm ignoring some nuance in the notations - but it will serve.)
So by definition we have $\aleph_1<\aleph_2$.
However, you're presumably interested in examples of sets with cardinality $\aleph_1$ and $\aleph_2$ respectively. This gets nastier. The good news is that there are such examples: the set of countable ordinals is a set of size $\aleph_1$ (in fact, it is the ordinal $\omega_1$), and the set of ordinals of cardinality at most $\aleph_1$ is a set of size $\aleph_2$ (in fact, it is the ordinal $\omega_2$).
The bad news is that if you want examples that aren't built from ordinals, you're basically out of luck; for example, it is consistent that the set of real numbers has cardinality $\aleph_{17}$ or has cardinality $\aleph_1$, so we can't even say whether $\aleph_2$ is the cardinality of some set of reals!
Even for $\aleph_1$ - which we know is the cardinality of some set of reals - the situation is very murky. It is consistent with ZFC that there is no "easily definable" set of reals of cardinality exactly $\aleph_1$; and similar results hold for $\aleph_2$ under the appropriate nontriviality assumption $2^{\aleph_0}\ge\aleph_2$. Look up descriptive set theory for more detail.

Tl;dr: The clearest description of higher $\aleph$s you'll get is in terms of sets of ordinals, and the universe of $\aleph$s is built up inductively; any other type of representation (e.g. as sets of reals, etc.) will generally not be possible (or at least it will be consistent that no such representation exists).
