A thin or posetal category a category in which, given any two objects $x$ and $y$ and any two morphisms $f$ and $g$ from $x$ to $y$, the morphisms $f$ and $g$ are equal:

$x \underoverset{\quad g \quad}{f}{\rightrightarrows} y
\;\;\;\implies\;\;\;
f = g$

Properties

Up to isomorphism, a thin category is the same thing as a proset (a preordered set). Up to equivalence, a thin category is the same thing as a poset. So mostly we just talk about posets here, but some references want to distinguish these from thin categories. (It is really a question of whether you're working with strict categories, which are classified up to isomorphism, or categories as such, which are classified up to equivalence.)