These are really the same thing.
You can start with one of our old answers: http://van.physics.illinois.edu/qa/listing.php?id=1469
Say that you just have the one pure substance in a sealed container with fixed volume bigger than the liquid. If there is no gas above, any molecules that escape (via thermal energy) into that vacuum are very unlikely to return. The gas buildup will continue until the rate of molecules returning to the liquid from the gas equals the rate at which they hop out of the liquid. The pressure of the gas is called the equilibrium vapor pressure. It goes up as the substance is heated, primarily because the extra energy helps molecules escape from the liquid.
If instead you put the liquid in an atmosphere of other gases, it will gradually evaporate at any temperature, since the escaping molecules have a huge space in which to wander off. However, if you look at what happens to a vapor bubble inside the liquid, it's like the gas in the fixed-pressure case. Bubbles will keep growing if it's above the boiling temperature, and that's just what we call "boiling".
What if we replaced the top of the container with a fixed-pressure sliding piston? If its pressure were greater than the vapor pressure, it would just squash the gas back into the liquid. If its pressure were less than the vapor pressure, the gas would keep pushing it up, expanding until all the liquid is used up. The temperature where that happens is the boiling point for that substance at that pressure.
(published on 06/01/10)