What actually makes boiling happen? Boiling can be breezed over easily with a few rudimentary diagrams and a couple equations, but I seek a deeper explanation. 
The definition of boiling is that the vapor pressure in the liquid is equal to the vapor pressure of the air. This seems reasonable in open containers because when pressure exceeds 1 atm, then bubbles can form. However, consider a closed container at lets say 40 degrees celsius. This is too cold for boiling, but the vapor pressure of the liquid must be equal to the pressure exerted by the vapor (system will move until this equilibrium is reached). Why doesn't this water boil? The pressures are the same so bubbles can form and boiling can occur.
My fundamental misunderstanding of boiling at the molecular level leads to more related questions:
When heat is added, why does the temperature rise until boiling point and then all the energy goes towards breaking bonds?
In melting water, the vapor pressure of the water and solid are equal so they can both coexist. If this is the case, isn't this technically the triple point? because water, solid, and vapor are present? (This can't be of course because melting point is not equal to triple point).
 A: Boiling will occur when the equilibrium vapor pressure at the temperature of the liquid is equal to the total pressure of the system and heat is being provided (either externally, or by the sensible heat of the liquid itself).  The gas phase in contact with the liquid can be comprised of pure vapor (in which case the total pressure is essentially equal to the equilibrium vapor pressure), or, it can be comprised of a mixture of vapor and air (in which case the total pressure essentially equal to the sum of the equilibrium vapor pressure of the boiling substance plus the partial pressure of the air).  
At 40 C, it is still possible for water to boil in a container if the gas is evacuated to a pressure below the equilibrium vapor pressure at 40 C (a non-equilibrium situation). Boiling can continue until the total pressure in the head space rises to a value equal to the equilibrium vapor pressure.  As long as the partial pressure of the substance in the vapor and the total pressure (if air is present) is less than the equilibrium vapor pressure of water at the liquid temperature, boiling will continue.  The heat of vaporization can be supplied by the liquid itself, so the water temperature will be dropping (assuming heat is not being directly supplied).  Once the equilibrium vapor pressure at the liquid temperature falls below the total pressure, boiling will stop.  Once the partial pressure of the vapor in the gas phase becomes equal to the equilibrium vapor pressure, evaporation will stop.
With regard to the question about the melting point and the triple point, the melting point is very close to the triple point.  At the triple point, pure water is present in all three phases (no air present in the gas phase) and the total pressure is equal to the equilibrium vapor pressure of the liquid and solid.  At the melting point, air is present in the gas phase at 1 atm., and the total pressure of the system is thus essentially 1 atm.  The difference between the melting point and the triple point is only about 0.01 C.
A: I think the question is really pertinent. If you heat the liquid without any impurity and in a flask with very smooth walls, kept at higher pressure until the temperature is just below the boiling point at that pressure and you slowly start to reduce the pressure, you will find that the water is not started to boil, this state is known as super heated state. If you drop a very small particle in that super heated water you will find a trail of bubbles behind that particle. 
This experiment actually gives a deeper understanding of boiling. At boiling point although the vapor pressure inside the liquid is equal to that of surrounding but the act of boiling comes from certain instabilities present in the liquid in the form of impurities, temperature hot spots etc. these instability create a small bubble at the bottom which comes up and grow in size and you will see random bubbles coming out of the water.
As far as second part of your question is concerned @chester miller has given a correct description of that. I would like to add that you have to provide enough heat to keep water at constant temperature and keep pumping the system to maintain it at boiling point otherwise your water will soon turn into ice (I have seen this experimentally).
