Hi! I'm going to take this one question at a time:

First question: Vapour pressure is the partial pressure of a gas when it's at equilibrium with a bulk liquid phase. To internalize what that means, think about the following:

You have a sealed container with only water in it. You pump away any gas, so that it's just water, and a vacuum inside this container. Some water is going to evaporate: after all, puddles still evaporate at 25 °C! At the beginning, water evaporates, and practically no condensation occurs, because there isn't any water vapour to condense. As more evaporation occurs, condensation starts happening, and those two rates are eventually equal. The system has reached equilibrium.

The pressure of the gaseous water, when it reaches equilibrium, in this example is the vapour pressure of water (at whatever temperature we used).

Second question: when the vapour pressure of a liquid is equal to the surrounding pressure, the liquid boils. Thus, there are two ways of making something boil: change its vapour pressure (by changing its temperature) or change the external pressure (by, say, climbing a mountain).

Third question: a pressure cooker works by increasing the pressure of the system. At 1 atm, water boils at around 100 °C. A pressure cooker maintain a pressure of about 2 atm. This means that in order to make water boil, you'll need to heat it up hotter than usual: that temperature winds up being about 121 °C. Since boiling water in the pressure cooker is about 20 °C hotter than usual, things will cook faster in it.

If anything's still fuzzy, or if any other questions come to mind, please don't hesitate to ask!
Mike