In the symplectic topology view on Gromov-Witten-Invariants some authors use what they call a Kuranishi structure on the moduli of stable maps. These were introduced by Fukaya and Ono and are also used in their big book on Fukaya categories.They are also used a lot in recent papers by Joyce.
The key feature of the Kuranishi structure is that there locally exists a homeomorphism or a diffeomorphism, depending on whether you follow Fukaya or Joyce, to a zero set of a section of the "obstruction bundle".
On the algebraic geometry side there also is something involving the term obstruction, namely the perfect obstruction theory on the moduli space of maps. This is a morphism [E-1 -> E0] -> LX, where L is the cotangent complex.
Here's my question: To what does the obstruction bundle on the symplectic topology side correspond on the algebraic side?
There are three candidates I can think of:
the kernel of E-1 -> E0
the thing the kernel of E-1 -> E0 maps surjectively to, mostly called the T2
The definition of the obstruction bundle on the symplectic side is a finite dimensional supspace of the cokernel of the linearized operator of the pseudoholomorphic curve equation, p.978 of Fukaya-Ono:"Arnold Conjecture and Gromov Witten Invariant." As a die hard algebraic geometer, that's just to hard to digest...