I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction theories have been used in algebraic geometry. The applications I know of are in Artin's criteria for showing that stacks are algebraic, and in the construction of virtual cycle classes in Gromov-Witten, Donaldson-Thomas, and similar theories.
Morally, an obstruction theory is supposed to control infinitesimal liftings: suppose $X \rightarrow Y$ is a map of algebraic geometry objects (schemes or stacks or whatever) and $S \subset S'$ is a square-zero extension of $Y$-schemes. An obstruction theory for $X$ over $Y$ is, vaguely speaking, a way of associating to any $Y$-morphism $S \rightarrow X$ an obstruction to the existence of an extension of that map to a $Y$-morphism $S' \rightarrow X$. In any individual lifting problem, an obstruction can usually be found in some cohomology group or other, but it is useful for some purposes (like the ones noted above) to have an abstract definition.
I know of four attempts to axiomatize the notion of an obstruction theory in algebraic geometry already:
1) Artin, M. Versal deformations and algebraic stacks
2) Fantechi, B. and Manetti, M. Obstruction calculus of functors of Artin rings, I
3) Li, J. and Tian, G. Virtual moduli cycles and Gromov--Witten invariants of algebraic varieties
4) Behrend, K. and Fantechi, B. The intrinsic normal cone