W(0) = 0; increments are independent; W(t) - W(s) is normal with mean 0 and variance t - s; paths are continuous.
The properties that matter: variance grows linearly in time, so standard deviation grows like sqrt(t). Paths are continuous everywhere but differentiable nowhere, which is why ordinary calculus fails and Ito calculus exists.
It is a martingale and a Markov process.
Geometric Brownian motion - the standard price model - exponentiates it, giving lognormal prices that stay positive.