  It is well-known that every first-order property on words is expressible
using at most three variables. The subclass of properties expressible with only
two variables is also quite interesting and well-studied. We prove precise
structure theorems that characterize the exact expressive power of first-order
logic with two variables on words. Our results apply to both the case with and
without a successor relation. For both languages, our structure theorems show
exactly what is expressible using a given quantifier depth, n, and using m
blocks of alternating quantifiers, for any m \leq n. Using these
characterizations, we prove, among other results, that there is a strict
hierarchy of alternating quantifiers for both languages. The question whether
there was such a hierarchy had been completely open. As another consequence of
our structural results, we show that satisfiability for first-order logic with
two variables without successor, which is NEXP-complete in general, becomes
NP-complete once we only consider alphabets of a bounded size.
