For example, if $H$ is a hyperplane, then $\mathbb{P}^n - H = \mathbb{A}^n$, which is a vector space.

If $n = m^2 - 1$, then we can regard $\mathbb{A}^{n+1}$ as the space of $m \times m$ matrices and take the hypersurface $H$ in $\mathbb{P}^n$ corresponding to the singular matrices. The complement $\mathbb{P}^n - H$ is $\mathbf{PGL}_n$.

If we restrict ourselves to irreducible $H$, are there any more examples besides the two above?

If we allow reducible hypersurfaces, then we can get a few more. We can realize the multiplicative group $\mathbb{G}_m$ as $\mathbb{P}^1$ minus two points, and removing the union of two distinct lines from $\mathbb{P}^2$ will give us $\mathbb{G}_m \times \mathbb{A}^1$. What can we say about the situation here?

The complement of a hypersurface is affine, so only linear algebraic groups will arise.

I haven't put much thought into the base field, so we can just start with $\mathbb{C}$.