The next one after $\aleph_2$ is $\aleph_3$. There's also $c$, the cardinality of the reals. In fact $c=\aleph_\alpha$ for some $\alpha$, but nobody knows which one...
Regading your new request for "tangible examples": In fact $\aleph_0$ is the cardinality of the integers, which I gather is what you mean by a tangible example. But no, $\aleph_1$ is not the carinality of the reals.
So what are these aleph things? One can prove that given a cardinal there is a smallest larger cardinal. So $\aleph_1$ is by definition the smmallest uncountale cardinal, $\aleph_2$ is by definition the smallest cardinal larger than $\aleph_1$, etc. That's as "tangible" as it gets.
Finally, regarding "Natural numbers, Real numbers, and the number of curves that can pass through a point?": Although you didn't actually say so, this sounds like you think the cardinality of that set of curves is larger then the reals. This is not so. If a "curve" is the graph of a continuous function, then the cardinality of the set of all curves, passing through a given point or not, is just $c$.