George Lowther at Almost Sure has written a terrific series of posts explaining stochastic processes and the stochastic calculus. Stochastic calculus is widely used in physics and finance, so there are many informal introductions that get across the main ideas in a form sufficient for applications. Most of the formal presentations of the subject seem very far away from the informal ones, to an unusual extent. For example, for the important technical notion of semimartingale the Wikipedia definition is the usual one, which has a very different flavor from the naive picture useful in applications. Lowther introduces it directly in terms of the stochastic integral, and the stochastic integral itself is introduced as a limiting process of random sums of a particularly simple form. The random sums are pretty much the same things you would write down in a naive presentation.