Serre's criterion for normality, the valuative criterion for normality, normality vs. S2, maybe even seminormality.
Added by request: here's how I think about Serre's criterion. Call a rational function pretty good if it doesn't blow up in codim 1. Call it very good if it's actually well-defined in codim 1. Then a normal space is one for which pretty good rational functions are actually functions, whereas an S2 space only asks that very good rational functions are actually functions. To see the difference, look at x/(x+y) on {xy=0}, to see that the latter is not normal despite being S2. So how can normality fail -- how can f's value be ambiguous in codim 1? If there are 2 ways to approach some divisor -- non-R1ness.
