From a number theoretic perspective, there are a few famous problems related to ranks of elliptic curves, which a lot of modern research in the area is geared towards solving. For example, Manjul Bhargava recently received the Fields medal partly for his work on bounding average ranks of elliptic curves (and proving that the Birch and Swinnerton Dyer conjecture is true for ever-increasing percentages of elliptic curves).

To describe some of the results: an elliptic curve over $\mathbb{Q}$ is a rational smooth genus 1 projective curve with a rational point, or in less scary terms, the set of solutions to an equation that looks like
$$E(\mathbb{Q}) = \{(x,y) \in \mathbb{Q}^2: y^2 = x^3 + ax + b\}$$
where $a, b \in \mathbb{Q}$. It's a fact that any such set forms a finitely generated abelian group, so by the structure theorem for such objects the group of rational points is
$$E(\mathbb{Q}) \cong \mathbb{Z}^r + \Delta,$$
where $\Delta$ is some finite group. Now, we have complete descriptions of what this group $\Delta$ can be - a Theorem of Mazur limits it to a small finite list of finite groups of size less than 12. However the values of $r$ are much more mysterious. We define the *rank of $E$* to be this $r = r(E)$.

Now, we know quite a lot about $r$ - for example, in "100%" of cases the rank is $0$ or $1$ (where here "100%" is used in the probablistic sense, not to mean that every elliptic curve has rank $0$ or $1$!). There is also the *Birch and Swinnerton Dyer Conjecture (BSD)*, which is one of the very open problems that you mention that nobody has any idea how to prove, but which most people believe. It relates the rank of the elliptic curve to the order of vanishing of its $L$-function at 1. Perhaps the strongest heuristic for it is that it's been proved in certain special cases, as well as Bhargava's work. So much of modern number theory research goes towards BSD, and it's one of the famous Millenium problems.

However, what we *don't* have much intuition with is:

**Question:** Are the ranks of elliptic curves over $\mathbb{Q}$ bounded? That is, is there some $R$ such that for any elliptic curve $E/\mathbb{Q}$, we have $r(E) \leq R$?

As of last year, it was very open - there were loose heuristics both ways. The largest rank we've found so far is a curve with rank at least 28, due to Elkies, which has been the record-holder for a long time now. As I mentioned before, Bhargava has proved the *average* rank is bounded by at least 1.5, and this was enough to win a Fields medal.

However, having said all that, I think there has been some excitement recently with some stronger heuristics that lean towards the rank being bounded. I don't know enough about these heuristics to comment any further, but there's more information here: http://quomodocumque.wordpress.com/2014/07/20/are-ranks-bounded/