You are dealing with "equilibrium" between the liquid and vapor phases of a pure single component (water). Several things should have been clearly stated in your text, or in the lecture material, but it appears that they may have been omitted --- I'll restate them here for you:
1) the Gibbs phase rule states that the number of degrees of freedom of a system is equal to the number of components plus two minus the number of phases present in the system, f = c + 2 - p ; what this means is that the number of independent variables (pressure, temperature, volume, composition, etc.) you have to play with, "degrees of freedom," is fixed; for this case with one component and two phases, it equals 1 (fix T and everything else is fixed, or fix P and T is fixed);
2) evaporation of a pure component (liquid, in this case) proceeds at constant T if P is fixed, or at constant P if T is fixed --- if your system is completely in the liquid phase at fixed T, and you gradually reduce pressure, the point at which the very first vapor forms is called the bubble point pressure; if you continue to maintain T, the system can then be expanded at constant pressure, the vapor pressure for the liquid, at the fixed T, until the last "drop" of liquid evaporates --- the only thing changing is volume and heat content; enthalpy of vaporization (per gram, per mole, per unit volume of liquid evaporated) at fixed T is constant, no matter what fraction of the liquid has been evaporated (I think this is your question in your last post);
3) the locii of points at which two phases exist in equilibrium for a single, pure liquid-vapor system plotted on a P,T diagram is the "liquid-vapor coexistence line/curve," or the "vapor pressure curve;" the properties (densities, heat capacities, etc.) of the liquid and vapor phases are different except at the high T,P end of this curve (the critical point);
4) "equilibrium" means that a system is in mechanical, chemical, and thermal equilibrium with itself and the surroundings --- "boiling" processes are not strictly speaking at equilibrium, but introductory courses seldom make such distinctions, and you have not mentioned any remarks in the problem about effects (superheating) that suggest that you are to consider this as anything other than an equilibrium state; that is, if the problem states "boiling water at some temperature T," it means that the liquid phase is in equilibrium with the vapor phase.
You have my apologies for my failure to make this easily translatable --- I tried, but I was more concerned with stating things correctly without getting into too many of the complications that go along with real processes.
Holler if the slightest thing is unclear to you --- I hope this comes a little closer to addressing your questions.
