 Solution

An $n$-type over a first-order theory $T$ is a set of logical formulas in the displayed $n$ free variables first-order consistent with $T$. A complete type chooses exactly one of $\varphi$ and $\neg \varphi$ for each such logical formula. Equivalently it is a maximal consistent such set. For parameter-free partial types the logical formulas use the original first-order language; partial types over parameters use additional names for those parameters.

A partial type $p(\vec{x})$ is principal if some logical formula $\theta (\vec{x})$ first-order consistent with $T$ implies every member of $p$ modulo $T$:Such a $\theta$ is a principal formula of a type of the partial type. For a complete type it isolates that partial type, so principal complete types are exactly isolated types.

For a tuple $\vec{a}$ in a first-order structure $\mathcal{A}$, its parameter-free partial type isIt is a complete type over $\mathrm{Th}(\mathcal{A})$ because the first-order structure decides the truth of every logical formula at that tuple.