I think this is mentioned by Andre as one of the motivations for the theory of motives. There is also, I believe, a Park/IAS book covering something related to this.
Title: 13/2 ways of counting curves Authors: R. Pandharipande, R. P. Thomas Categories: math.AG Algebraic Geometry (math.SG Symplectic Geometry; physics.hep-th High Energy Physics - Theory) Comments: Small corrections. 50 pages, 4 figures. To appear in proceedings of “School on Moduli Spaces”, Isaac Newton Institute, Cambridge 2011 MSC: 14N35, 14N Abstract: In the past 20 years, compactifications of the families of curves in algebraic varieties X have been studied via stable maps, Hilbert schemes, stable pairs, unramified maps, and stable quotients. Each path leads to a different enumeration of curves. A common thread is the use of a 2-term deformation/obstruction theory to define a virtual fundamental class. The richest geometry occurs when X is a nonsingular projective variety of dimension 3. We survey here the 13/2 principal ways to count curves with special attention to the 3-fold case. The different theories are linked by a web of conjectural relationships which we highlight. Our goal is to provide a guide for graduate students looking for an elementary route into the subject.
nLab page on Enumerative algebraic geometry