What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?

The cardinals are the following ones: $$0,1,2,3,4,5,6,\dots,\aleph_0,\aleph_1,\aleph_2, \aleph_3,\dots,\aleph_\omega,\aleph_{\omega+1},\dots,\aleph_{\omega2},\aleph_{\omega2+1},\dots $$ Where $\aleph_0$ is the first infinite cardinal (the cardinality of each infinite countable set), so $\aleph_0\notin\mathbb N$, is not said to be an integer in the ordinary way. Then $\aleph_1$ is the next cardinal, and so on... (and this "and so on..." also includes some knowledge about the ordinals).

By Cantor's theorem ($|P(A)| > |A|$ for all sets $A$) we have that for every cardinal there is a bigger cardinal. By the well foundedness and the axiom of choice in ZFC, we have that every cardinal is a cardinal of a well-ordered set (which is in bijection to some ordinal), and it follows that there is always a next cardinal..

The cardinal number $\aleph_1$ is defined as the cardinality of the set of all countable ordinals, i.e. ordinals of cardinality $\le\aleph_0$. If you believe that that set of countable ordinals exists, then you've got it.

Perhaps you are asking for an example of a set of cardinality $\aleph_1$. Cantor thought the set of all real numbers would be an example. In fact, he thought he had a proof. Then he found a hole in the proof, and restated the question as a hypothesis, which has come down to us as "The Continuum Hypothesis": the cardinality of the reals is $\aleph_1$. No one has been able to prove this, nor to disprove it: In 1940, Gödel published his proof that this hypothesis cannot be disproved on the basis of the axioms of mathematics generally accepted at the time and, in 1963, Cohen published his proof that it cannot be proved either.

The bottom line is, no one has ever been able to present a set [EDIT: but see clarification in the comments] provably of cardinality $\aleph_1$ and, on our current understanding of these things, no one ever will.

If that doesn't answer your question, perhaps you'd like to edit your question so someone will be able to get what exactly you're asking.