Molarity of a compound is defined as number of particles per volume. Normality of a compound is context-dependent and is always some integer multiple of molarity, so that normality = x * molarity. The "x" depends on the context, but in general it tells what is the stoichiometry in some chemical reaction (if it goes to completion).
Using EDTA as an example, it has 4 acidic groups, so one EDTA molecule can lose 4 protons, and x = 4.
H4-EDTA <-> EDTA + 4H
When dealing with Calcium chelation, one EDTA molecule can only bind one Calcium, thus x = 1.
EDTA + Ca <-> Ca-EDTA
Hence, 1 mM (one millimolar) EDTA solution is 4 mN (four millinormal) in acid-base titration and 1 mN (one millinormal) in Calcium titration.
Actually, when you calculate reaction stoichiometries, you use normality all the time, it is just that when you do water analysis (I assume this has something to do about metal chelation), x = 1, and the normality and molarity have the same value.