2 Added connection to $B_3$

$\widetilde{\textit{SL}_2(\mathbb{R})}$ is not so complicated, but one of the best descriptions is just that. It has no faithful finite-dimensional representations, which makes things a little tricky. It's easy to see that $\textit{SL}_2(\mathbb{R})$ is topologically a solid torus, so it does have a central extension by $\mathbb{Z}$.

Another description of this central extension is to think of $PSL_2(\mathbb{R})$ acting on the hyperbolic plane. Then $\widetilde{SL_2(\mathbb{R})}$ is pairs $(g,\phi)$ where $g \in PSL_2(\mathbb{R})$ and $\phi:\mathbb{R}\to\mathbb{R})$ is a map that induces the map from $S^1$ to $S^1$ induced by $g$ on the circle at infinity. (Note that this goes by a quotient from $SL_2(\mathbb{R})$ and then centrally extends back again.)

For the relation to $B_3$, you can think of $PSL_2(\mathbb{Z})$ as the mapping class group of the 4-times punctured sphere, with one puncture (at infinity) distinguished and required to go to itself. To go from there to its central extension $B_3$, you remove a disk (rather than a point) around infinity, and only look at isotopies that fix that boundary, which is similar to what happened at the circle at infinity above.

More generally, all mapping class groups have a unique central extension; if there's a distinguished point, you can do it in the braid-like way (looking at isotopies that fix a boundary circle), while if there isn't, you do it by lifting the map of the surface to a map from $\mathbb{H}^2$ to itself (not an isometry) and then lifting the action on the circle at infinity to a map from $\mathbb{R}$ to itself. (These are equivalent when they are both defined.)

(I still feel like it ought to be possible to make the connection crisper.)

1

$\widetilde{\textit{SL}_2(\mathbb{R})}$ is not so complicated, but one of the best descriptions is just that. It has no faithful finite-dimensional representations, which makes things a little tricky. It's easy to see that $\textit{SL}_2(\mathbb{R})$ is topologically a solid torus, so it does have a central extension by $\mathbb{Z}$.

Another description of this central extension is to think of $PSL_2(\mathbb{R})$ acting on the hyperbolic plane. Then $\widetilde{SL_2(\mathbb{R})}$ is pairs $(g,\phi)$ where $g \in PSL_2(\mathbb{R})$ and $\phi:\mathbb{R}\to\mathbb{R})$ is a map that induces the map from $S^1$ to $S^1$ induced by $g$ on the circle at infinity.