What is this matrix/operator notation $\otimes$?

It's the Outer/Tensor product and defined by $u \otimes v = uv^\top$ or alternatively, you can think of it as the function

$$ (u \otimes v)(x) = \langle v, x \rangle u, $$

which is of course what you get when you multiply $uv^\top$ and $x$ together.

Note: there is also a Kronecker product which uses the same symbol and the relationship is

$$ u \otimes_{\rm out} v = u \otimes_{\rm kr} v^\top. $$

The Kronecker product of two column vectors $(a_i), (b_j)$ is a vector whose $(in + j)$-th entry is $a_ib_j$. The outer product is a matrix whose $(i,j)$-th entry is $a_ib_j$.