A 2-poset is any of several concepts that generalize posets one step in higher category theory. One does not usually hear about $2$-posets by themselves but instead as special cases of $2$-categories, such as the locally posetal ones.

$2$-posets can also be called (1,2)-categories, being a special case of (n,r)-categories. The concept generalizes to $n$-posets.

Definition

Fix a meaning of $\infty$-category, however weak or strict you wish. Then a $2$-poset is an $\infty$-category such that all parallel pairs of $j$-morphisms are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, not even whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than a locally posetal $2$-category for your preferred definition of $2$-category, but it will be equivalent if you ignore irrelevant data.

Just as the motivating example of a $2$-category is the $2$-category Cat of categories, so the motivating example of a $2$-poset is the $2$-poset Pos of posets.