What does $\langle dW_1, d W_2\rangle = \rho$ mean?
It shows up in stochastic differential equations involving brownian motions. I am told it's the "correlation", but why, how, and what does it mean?
In particular, I don't know what the scalar product of stuff like $"dW_1"$ is.
Given a continuous square integrable martingale $X_t$, there is a unique predictable process $\langle X\rangle_t$ such that $X^2_t-\langle X\rangle_t$ is a martingale. We call $\langle X\rangle$ the quadratic variation of $X$. If $X$ and $Y$ are two continuous square integrable martingales, then the quadratic covariation is defined as
$$\langle X,Y\rangle_t:=\tfrac14(\langle X+Y\rangle_t-\langle X-Y\rangle_t)$$
and you can easily check this is equal to $\langle X\rangle_t$ when $X=Y$. Then $\langle dX,dY\rangle:=d\langle X,Y\rangle$, that is, it is simply (Ito) integration with respect to the process $\langle X,Y\rangle$.