What is the 2D elliptical projection of the Celestial Sphere called, and how can I make one?
I want to say that's a Mollweide projection. I know that they're pretty common in astronomy; many images of the CMB use them. I was actually working with one recently. Given latitude $\varphi$ and longitude $\lambda$, the $x$ and $y$ coordinates of an object are
$$x=R\frac{2\sqrt{2}}{\pi}(\lambda-\lambda_0)\cos\theta,\quad y=R\sqrt{2}\sin\theta$$
for projecting from a sphere of radius $R$. $\theta$ is the solution to
$$2\theta+\sin2\theta=\pi\sin\varphi$$
Similar images can be found here (labeled as Mollweide projections), which are of the same shape and proportions:



Matplotlib will let you do Mollweide projections; a nice tutorial can be found here. Basemap is a possibility, although I haven't used it before. It has quite a few different built-in projections. There are some interesting features available in the different matplotlib implementations, and using the ephem package, you can use astronomical data. That said, I've found that the transformations aren't that hard to implement by hand; I was able to get some basic projections done from scratch.