I know this question is very old, but the answers here are unsatisfactory, and so for the benefit for people with same questions in the future, let me clarify important points. I have assumed ideal scenario neglecting inter molecular forces.
Consider an open container with liquid phase of a substance with no molecule in gas phase above it, only air. At the top surface, there would be invariably some molecules with high energy due to their random motion. Therefore, they would begin to escape the liquid since nothing prevents them to do so. This is called evaporation. The liquid would continue to evaporate till any more evaporation would lead to partial pressure of its gas phase get in liquid region. This gas pressure is known as vapor pressure of the liquid, and is equal to maximum pressure in which the substance can remain in gas phase at that temperature. Note that liquid at this point has pressure equal to total gas pressure above it (atmospheric pressure).
Take a control volume inside the liquid. Remember that due to thermal motion, the molecules inside the control volume try to expand into gas phase, but since the gas phase pressure at that temperature is lower than the pressure by liquid, it can't expand. But as soon as the pressure exerted by the liquid becomes equal to the gas pressure, the molecules inside can easily transform into gas phase. (Note that it is not that straightforward--some seeding is required for bubble to form due to surface tension forces while the surrounding liquid is superheated, but I digress.)
And what is the pressure of the liquid equal to? Atmospheric pressure. But only when the container is open. So when container is open, and either atmospheric pressure or the temperature of the liquid is brought such that the pressure of the liquid becomes equal to its vapor pressure, then the thing boils.
If the container is closed, and all the air is removed, the pressure inside the liquid is equal to vapor pressure, and there will always be vapor pressure present, even if in minuscule amount. (You can't have a container filled with only liquid. Think about that for a moment. Similarly you can't have a container filled with only ice. Some water, or water vapour, or even combination of the two will appear alongside.) And in that case, any change in conditions of pressure or temperature will be immediately reflected by boiling or condensation of the system. That is, a closed container with only water inside can boil at any temperature.
So in summary, vapor pressure is the maximum gas pressure possible at that temperature. The liquid always tries to expand and attain vapor pressure, and in case the inside pressure is greater than the vapor pressure, the inside of the liquid can't do so, otherwise it would boil. However, the surface of the liquid, due to imbalanced forces, can do it at any point by evaporating, and thus forming gas above it till vapor pressure is attained.