Molarity, moles of solute per liter of solution, is a convenient way to express concentration of solutions for chemical applications. That's because, in chemical calculations based on chemical reactions, we always must convert amounts of substances to moles. This is because a balanced chemical reaction is "in" moles by its very nature. When we balance a reaction we are finding the simplest whole number mole ratios between the reactant and product substances.
Since most reactions work best if solutions of reactants are mixed together (which achieves better mixing than if we mix the pure substances themselves without a solvent) expressing the concentration of the solutions in mole units is helpful. Knowing the molarity of a solution, we can calculate the number of moles of the solute in that solution in any volume of it.
Why use volume instead of mass in the denominator? I think simply because it's quicker and easier in the lab to measure and dispense volumes of solutions in glassware such as graduated cylinders instead of weighing each substance on a balance. The convenience factor again. There are concentration units that do use mass units in the denominator, such as molality (moles of solute per kg of solvent) and percent by mass (grams of solute per total grams of solution, times 100).
So remember,
molarity, M = moles of solute
liter of solution
and cross-multiplying gives
moles of solute = volume in liters X molarity 
Steve
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