In Martin Ziegler: A Course in Model Theory it is stated on page 53 that
Definition 4.3.8. A theory $T$ is small if $S_n(T)$ are at most countable for all $n<\omega$.
And it is followed by these two statements I don't understand:
A countable theory with at most countably many non-isomorphic at most countable models is always small. The converse is not true.
I cannot see a direct connection between isomorphism classes of models of $T$ and types. (Something like: realisation sets $\leftrightarrow$ clopen sets $\leftrightarrow$ types?)