Roughly speaking, two categories $\mathcal{C}$ and $\mathcal{D}$ are isomorphic if there are functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ so that $F\circ G$ is the identity functor on $\mathcal{D}$ and $G\circ F$ is the identity functor on $\mathcal{C}$. (See here.)
I'm interested in a similar relation between categories that represent partially-ordered sets where $F\circ G$ is no greater than the identity functor on $\mathcal{D}$, and $G\circ F$ is no smaller than the identity functor on $\mathcal{C}$. Is this situation well-known? Does it have a name?
* I'm a complete newbie to category theory. Please take this into account when answering the question.
What you describe is precisely an adjunction between two posets. It's more commonly called a (monotone) Galois connection.
Indeed, the usual definition of an adjunction is a pair of functors $F,G$ as in your question, together with natural transformations $\eta:I_{\mathcal{C}}\Rightarrow G\circ F$ (the unit) and $\epsilon:F\circ G\Rightarrow I_{\mathcal{D}}$ satisfying the so-called "triangular identities" $G(\epsilon)\circ \eta_G=1_{G}$ and $\epsilon_F\circ F(\eta)=1_{F}$. But when the two categories are preorders, these identities are trivially satisfied, and the existence of natural transformations amounts to the conditions $c\leq GF(c)$ and $FG(d)\leq d$ for all $c\in \mathcal{C}$ and $d\in \mathcal{D}$. The bijection $\mathcal{C}(c,G(d))\cong \mathcal{D}(F(c),d)$ then tells you that $c\leq G(d)$ if and only if $F(c)\leq d$.