Wat is Brownian motion, Gaussian process?

A Gaussian process is a process $(X_t)$ whose finite-dimensional laws $(X_{t_1}, \dots, X_{t_k})$ are all multivariate normal; its law is fixed by its mean $m(t)$ and covariance $c(s,t)$. A (standard) Brownian motion $(W_t)_{t \ge 0}$ is an adapted process with $W_0 = 0$, continuous paths, and increments $W_t - W_s$ independent of $\mathcal F_s$ and distributed $\mathcal N(0, t - s)$ for $s < t$.