In general, taking the non-relativistic limit for a moving particle is to assume that its velocity is much smaller than the speed of light, i.e. that $$\frac{v}{c}\ll 1.$$
In this limit, the laws of Special Relativity coincide with the laws of Newtonian physics, and (most) relativistic effects can be ignored. For example, in Special Relativity the equations that govern the transformations from one inertial frame $S$ to another $S'$ that's moving with a velocity $v$ with respect to $S$ are the Lorentz Transformations:
\begin{aligned}
x' &= \gamma \left( x - v t \right),\\
t' &= \gamma \left( t - \frac{v}{c^2} x \right),
\end{aligned}
where $$\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}.$$
In the limit $v/c \ll 1$, $\gamma = 1$, and you can see that these just reduce to the Galilean Transformations of Newtonian physics:
\begin{aligned}
x' &= x - vt\\
t' &= t
\end{aligned}
In the document you've linked, the author speaks about temperature since at thermal equilibrium the "temperature" of a substance is proportional to the average kinetic energy of the molecules that compose it. Therefore, as the "soup" that the document speaks of cools, the average kinetic energy of its constituents decreases. As a result, the velocities of these constituents start to decrease, meaning they become "less" relativistic.
There is, however, no strict threshold beyond which a particle is "non-relativistic". That being said, there is another way to roughly quantify this: the energy of a (massive) particle in Special Relativity is given by: $$E = \gamma m c^2 \approx m c^2 + \frac{1}{2} m v^2 + ...$$
The first term in the expansion above is the rest mass (energy) and the second is the "kinetic" energy. If a particle's total kinetic energy is much larger than its rest mass energy, you should be able to see that this means the particle is "relativistic".
Given that the average kinetic energy of a system of particles is simply related to the temperature, this basically means that (in units where $c = 1$ and the Boltzmann constant $k_B = 1$) when $T \gg m$, a massive particle can be considered to be relativistic, and when $T \ll m$ a massive particle can be considered to be non-relativistic. In other words, as a system of particles cools, $T \sim m$ is a rough threshold for the particle to be "non-relativistic".
