Recall the following
Fact 1. The equation
$$
a x + b y = c
$$
has no solution with $x,y \in \Bbb{Z}$ if $(a,b) \nmid c$, while it admits infinitely many if $d = (a,b) \mid c$. Further, in this case the general solution (i.e. every solution) is of the form
$$
x = \frac{b}{d} k + x_0 \quad y = -\frac{a}{d} k + y_0
$$
for any integer $k$, where $(x_0,y_0)$ is a particular solution (e.g. any one given or known solution).
Note that you can rewrite your equation as
$$
2 x_1 + 12 x_2 + 3 x_3 = 2 x_1 + 3 ( 4x_2 + x_3) = 7 \label{eq:orig} \tag{1}
$$
In particular, every solution $(x_1,x_2,x_3)$ of \eqref{eq:orig} gives you a solution $(x_1, y = 4x_2 + x_3)$ of
$$
2 x_1 + 3 y = 7 \label{eq:partial_1} \tag{2}
$$
thus we will first solve this other equation and then for each of these solutions we will look for the corresponding solution of \eqref{eq:orig}, if there are any.
Observe that $(2,3) = 1$, thus \eqref{eq:partial_1} has infinitely many solutions. Furtehrmore, $x_1 = 2, y = 1$ is a particular solution of \eqref{eq:partial_1}, hence from Fact 1 you know that every solution of \eqref{eq:partial_1} is of the form
$$
x_1 = 3s + 2 \quad y = -2s + 1 \qquad \forall s \in \Bbb{Z}
$$
Now, for any fixed $s$ we go on and solve the equation
$$
4 x_2 + x_3 = -2s + 1 \label{eq:partial_2} \tag{3}
$$
Again, observe that $(4,1) = 1$ and that $x_2 = -s$, $x_3 = 2s + 1$ is a particular solution of \eqref{eq:partial_2}. Hence it follows from Fact 1 that the general solution is of the form
$$
x_2 = t - s \quad x_3 = -4t + 2s + 1 \qquad \forall t \in \Bbb{Z}
$$
Putting it together, we conclude that the general solution of \eqref{eq:orig} is of the form
$$
x_1 = 3s + 2 \quad x_2 = t - s \quad x_3 = -4t + 2s + 1 \qquad \forall s,t \in \Bbb{Z}
$$
