Here is one example that I always found fascinating.

• Frames are complete lattices which satisfy the infinite distributive law $$U \wedge \bigvee_{i \in I} V_i = \bigvee_{i \in I} U \wedge V_i.$$ In pointfree topology, these are used to abstract the lattice of open sets of a topological space.

• Complete Heyting algebras are complete lattices which have a binary operation ${\Rightarrow}$ that satisfies $$U \wedge V \leq W \quad\mbox{iff}\quad U \leq V \Rightarrow W.$$ These are primarily used to interpret intuitionistic logic.

The fact that these two types of lattices are cryptomorphic is essentially the Adjoint Functor Theorem (when viewing the underlying partial order as a category).