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

$\bar \partial \varphi + \frac{1}{2}[\varphi, \varphi]=0$,

were $\varphi \in \mathcal{E}^{0,1}(T^{1,0})$. In order to do this, one first look at a solution which is a formal power series

$\varphi(t)=\varphi_1 t + \varphi_2 t^2 + \varphi_3 t^3 +...$

Collecting powers of $t$ we obtain equations

$\bar \partial \varphi_1=0$

$\bar \partial \varphi_2 + \frac{1}{2}[\varphi_1, \varphi_1]=0$

...

The first equation states that $\varphi_1$ is an harmonic form, that is an element of
$\mathcal{H}^1(T^{1,0})$. By Hodge Theorem, this space can be identified with $H^1(X, T_X)$, which is exactly the space parametrizing "first-order" deformations.

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 $[\varphi_1, \varphi_1]$ is a coboundary. So the class of $[\varphi_1, \varphi_1]$ in $H^2(X, T_X)$ 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 Spec $k[\epsilon]/(\epsilon^k)$.