Why is boiling not just a surface phenomenon? The boiling point of a liquid is defined as

The temperature at which the vapor pressure of the liquid is equal to the external atmospheric pressure.

That being the case, why, in a pot of boiling water, does the water have a pretty stable liquid-air interface? Why are bubbles seen as coming from inside the body of water if boiling is essentially defined as a surface phenomenon?
 A: The quoted definition may sound confusing because of the use of "vapour pressure", which is not necessarily related to the liquid-air surface. This can be made clearer by a different example:
Suppose you have a water container at room temperature and you heat only a small volume at the centre of the container (by using focused radiation of some sort, for example). That heated volume would be considered as "boiling" when its phase changes to gas phase (due to the elevated temperature), and the pressure of that gas is higher than the surrounding pressure (hydrostatic + atmospheric).
In this example, "vapour pressure" is the (elevated) pressure of the heated volume, and "external pressure" is the sum of hydrostatic and atmospheric pressures.
Also, as @brandon-enright mentioned, 

That's why you don't see boiling even around thermal vents in the
  bottom of the ocean. The water just superheats.

So boiling is not a surface phenomena, as it can happen far away from any interface and quench far before reaching any interface.
A: If the pressure in the entire pot of water is less than the vapor pressure of boiling water, and substantially equal to the ambient atmospheric pressure, then the entire pot of water will begin to vaporize, including its interior which vaporizes into bubbles.  Bubbles form because the ambient pressure surrounding them is less than the vapor pressure of water at the boiling temperature.  As Brandon Enright indicates in his comments, boiling is an indication that the vapor pressure of the water exceeds the ambient pressure.
A: Boiling is clearly not a surface phenomenon. But vaporising is.
Boiling happens at all the points inside the liquid whereas when vaporising only the molecules at the surface escape into the space above. 
And it is true that a liquid boils when its saturated vapour pressure equals external (room) pressure. But it is not to be confused with vaporising. Boiling corresponds to a phase change. 
The bubbles are due to the liquid being converted into gas. And as the liquid is heated from the bottom (like from a bunsen burner) the gaseous product formed rises to the top due to its low density. 
