If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equation
$\partial \bar \varphi + \frac{1}{2}[\varphi, \varphi]=0$,
were $\varphi \in H^1(X, T_X)$, identified with the space of harmonic 1-forms. In order to do this, one first look at a solution which is a formal power series
$\varphi(t)= t \varphi_1 + t^2 \varphi_2 + t^3 \varphi_3+...$
Collecting powers of $t$ we obtain equations
$\partial \bar \varphi_1=0$
$\partial \bar \varphi_2 + \frac{1}{2}[\varphi_1, \varphi_1]=0$
...
The first equation states that $\varphi_1$ is an harmonic form, according to the fact that the "first-order" deformations are parametrized by $H^1(X, T_X)$.
The second equation states that you can extend the first order deformation to a second-order one (i.e., you can solve the Maurer-Cartan equation modulo $(t^3)$ ) if and only if the 2-cocycle $[\varphi_1, \varphi_1] \in H^2(X, T_X)$ is a coboundary. So $[\varphi_1, \varphi_1]$ is the "primary obstruction" to your deformation problem.
In this way, you can try to solve modulo higher and higher powers of $t$. If all the higher order obstructions vanish and the series defining $\varphi(t)$ converges, you obtain a "genuine" deformation, namely a deformation over a small disk.
Now it should be clear that, in order to generalize this in the algebraic framework, you need a substitute for the step "solve the Maurer-Cartan equation modulo $(t^k)$ ". This substitute is roughly speaking obtained by considering deformations over $\mathbb{C}[\epsilon]/(\epsilon^k)$.