A normal mode is a specific type of vibration that a molecule can exhibit. They are "normal" because each type of vibration is distinct and cannot be replicated by any combination of the other types. Any possible type of vibration that a molecule can exhibit is just a combination of its normal modes.
The numbers can probably be derived from symmetry arguments, I don't know how, but it's easier just to draw out molecules for n = 2 and 3 and try to convince yourself that there is an agreement with the formulas.
There are loads, Bernath (2005) is pretty good for a conceptual overview in the context of spectroscopy, should be in most scientific libraries. But if you're serious about molecular spectroscopy.... Penner or Herzberg are solid, detailed (if a little old) texts.