Einstein gave an axiomatic definition of Brownian motion, in use in mathematical physics, statistical mechanics and probability theory:
(1-dimensional) Brownian motion is the probability measure on the set of continuous functions B: R -> R such that
  1. For all nonintersecting intervals (a,b) and (c,d), the random variables B(b)-B(a) and B(d-c) are independent.
  2. For all intervals (a,b), the variable B(b)-B(a) has a normal distribution with variance b-a.
This captures our intuitions of what a "random" motion of a particle should be. Unfortunately, he forgot to prove that such a distribution exists!

Actual constructions (obviously only in the sense of mathematical analysis, not in the sense of mathematical logic, if you're into that sort of thing) only came later; Paul Lévy and Wiener provided correct constructions.

In fact, even slight changes to the definition can yield nonexistent results! For instance, Durrett provides an example where replacing the normal distribution by a Cauchy distribution can be proven not to exist: attempting to follow through the construction yields a function that is almost always nowhere continuous.

To get multidimensional Brownian motion, it is sufficient to take independent Brownian motions along each axis; the unique property of the multidimensional normal distribution show that this is a good idea, and the result independent of the particular choice of axes.

Brownian motion is a very important example of a continuous stochastic process. Obviously, it's a continuous martingale.

Stochastic integrals are an attempt to define an integral with respect to the "measure" dB(s), in the sense of Stieltjes integrals.