I can't remember the nomenclature (if it exists) for a poset with the property in the title, that is, a poset $(P,\leq)$ with the following property:
If $x<y$ in $P$, then there exists $z\in P$ with $x<z<y$.
I remember that a nice application of this property is in characterizing the unique countable homogeneous total order, i.e., any countable order with this property and no minimum nor maximum is isomorphic (in the order-theoretic sense) to $(\mathbb{Q},\leq)$, but I couldn't find the nomenclature.
Also I'm not very sure wheter I should ask for a total order or not, but you can correct me if that's the case.