The term first appeared in Chapter 3 of Lurie's HTT, p144. Where he says
The simplicial category $Cat^\Delta_\infty$ has as objects (small) $\infty$-categories.
What are small $\infty$-categories? what are $\kappa$-small $\infty$-categories?
It seems that we first define a universe $U$ for which these "small" objects live in. Then construct the universes $U(\kappa)$ with respect to each **-cardinal.
I am unclear how to make this precise. (i.e. what to put in **).
As Lurie mentions in section 1.2.15, he works in the framework of Grothendieck universes for convenience. In particular, with some inaccessible cardinal $\kappa$ fixed in the background, a small set (or $\kappa$-small set to avoid ambiguity if $\kappa$ might vary) is an element of the Grothendieck universe $U(\kappa)$ of sets of rank less than $\kappa$ (or as a set theorist would call it, $V_\kappa$). So, a small $\infty$-category is just an $\infty$-category that is an element of $U(\kappa)$.
Concretely, up to isomorphism, a small $\infty$-category is just an $\infty$-category $X$ such that $X_n$ is a set of cardinality less than $\kappa$ for each $n$.