We obtain explicit formulas for the number of non-isomorphic
elliptic curves with a given group structure (considered as an abstract abelian group).
Moreover, we give explicit formulas for the number of distinct group structures of all
elliptic curves over a finite field. We use these formulas to derive
some asymptotic estimates and tight upper and lower bounds for
various counting functions related to classification of elliptic
curves accordingly to their group structure. Finally, we present
results of some numerical tests which exhibit several interesting
phenomena in the distribution of group structures.