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?)
Every $n$-type (over a countable language) is realized in some countable model, by Lowenheim-Skolem. So the number of $n$-types extending $T$ is at most the number of $n$-tuples of elements of countable models of $T$ up to isomorphism. Each countable model has only countably many $n$-tuples of elements, so if there are only countably many countable models up to isomorphism, then $S_n(T)$ is countable.