With respect to acid-base chemistry, normality is a measure of the concentration of H+ or OH- (or equivalent) that can be produced by a particular acid or base. For most it is equal to the molarity, which is simply the concentration. A good example of when it's not equivalent is with diprotic and triprotic acids.
For instance with sulfuric acid (H2SO4), which is diprotic, a 1M solution would be 2N. In this case 1M = 2N. That is because a single molecule of (H2SO4) can produce two H+.
A single molecule of NaOH produces a single OH- anion, so for NaOH, M = N always.