Boiling is not well described? It is said that boiling occurs only when the vapour pressure becomes equal to the pressure surrounding the vapour (say atmospheric pressure). 
Why does it take place only satisfying these conditions ? They say that it is only in this condition that equilibrium between phases takes place. 
If so, doesn't the continued heat supply cause a change in the equilibrium ?? 
 A: The issue is not "equilibrium" - the issue is boiling.
During boiling, there is explicitly NO equilibrium: the water wants to get out of the liquid phase, and into the vapor phase. The temperature of the liquid is sufficiently high that liquid can evaporate below the surface (strictly speaking the temperature must be slightly above boiling for that, as the pressure below the surface is above ambient pressure) and form bubbles. As these bubbles rise, they expand - and at the surface, they release their vapor into the atmosphere.
There is no "equilibrium" here - as the liquid continues to be heated, it continues to evaporate. And unless the atmosphere above the liquid is emptied of all species other than water vapor, the process will continue more in the forward (evaporation) direction than the reverse (condensation) direction: the definition of "not in equilibrium".
At temperatures below boiling, the rate at which liquid escapes and becomes vapor ultimately becomes the same as the rate at which vapor condenses back into the liquid. That's when equilibrium is possible - although not always attained. It depends on the degree of circulation of (dry) air above the liquid.
So I would say you were right - your question

doesn't the continued heat supply cause a change in the equilibrium

can be answered with "YES". There is no equilibrium during boiling.
