What is so special about elliptic curves? There seems to be sources like this, this also, and some introductions that discuss elliptic curves in general and how they're used.  But what I'd like to know is why these particular curves are so important in cryptography as opposed to, let's say, any other polynomial degree $\gt$ 2 which you can then mod over some group.  It seems like once a modulus is applied then other function types should be acceptable as well.

It seems even less intuitive when just looking at the bubble vs curve as here:
 

Since there are other curves (let's say anything from a sin wave to the $x^3 + x$ or even just some unusually shaped contour) that could do the job.  It seems like they would provide much more surface area to get a larger space in $\mathbb{Z}_p$ or really just more possible combinations of connecting lines from some arbitrary $P$ and $Q$ to get $R$ as opposed to something as restrictive (on the graph) as beginning from some bubble (which would seem to unnecessarily reduce the possible combinations) and then use a modulus to implement the discrete logarithm problem.

Sorry if this seems a little naive of a question, I'm trying to write an implementation right now and just to understand it fully even if that means asking something that is taken for granted.  Perhaps just walking through a simple example (most of the ones I've searched are anything but), just a few sentences, would be rather helpful, from "A wants to talk to B" all the way up to "now E can't listen in between A and B".

EDIT:

So it seems like this is the version of elliptic curves over a finite field:



Yes that looks pretty random.  But I'm still not really seeing why they are the only equations that have cryptographic significance.  It's difficult to imagine that if you simply took some other higher degree equation and applied modulus (to place within a group), then it seems like it would make sense that you'd get something that's also comparatively random.
Elliptic curves are not the only curves that have groups structure, or uses in cryptography. But they hit the sweet spot between security and efficiency better than pretty much all others.

For example, conic sections (quadratic equations) do have a well-defined geometric addition law: given $P$ and $Q$, trace a line through them, and trace a parallel line that goes through the identity element. Here's a handy picture for one of the best known conics, the unit circle $x^2 + y^2 = 1$:



If you take the identity element to be $(1, 0)$, then you get the very simple addition formula (modulo your favorite prime)

$$ (x_3, y_3) = (x_1x_2 - y_1y_2, x_1y_2 + x_2y_1)$$

This is much faster than regular elliptic curve formulas, so why not use this? Well, the problem with conics is that the discrete logarithm in this group is no stronger than the discrete logarithm over the underlying field! So we would need very large keys, like prime-field discrete logarithms, without any advantage. That's not good.

So we move on to elliptic curves, which do not have reductions to the logarithm on the underlying field.

But wait, we can generalize elliptic curves to higher degrees. In fact, 

$$ y^2 = x^{2g+1} + \ldots $$

when $g &gt; 1$ and some restrictions are respected, is called a hyperelliptic curve, and we can work on it too. But for these curves there does not exist a nice geometric rule to add points, like in conics and elliptic curves. So we are forced to work in the Jacobian group of these curves, which is not the group of points anymore, but of divisors (which are kind of like polynomials of points, if that makes any sense). This group has size $\approx p^g$, when working modulo a prime $p$.

Hyperelliptic curves do have some advantages: since the group size is much larger than the prime, we can work modulo smaller primes for the same cryptographic strength. But ultimately, hyperelliptic curves fall prey to index calculus as well when $g$ starts to grow. In practice, only $g \in \{2,3\}$, i.e., polynomials of degree $5$ or $7$ offer similar security as elliptic curves. To add insult to injury, as Watson said, the addition formulas also get much more complicated as $g$ grows.

There are also further generalizations of hyperelliptic curves, like superelliptic curves, $C_{a,b}$ curves, and so on. But similar comments apply: they simply do not bring advantages in either speed or security over elliptic curves.