I like order theory very much, and am always excited to discover mathematical structures that can be intrinsically endowed with a partial order. "Easy" examples are $T_0$ topological spaces (with $x \leqslant y \Leftrightarrow x \in \overline{\{y \}}$) and vector spaces endowed with a pointed convex cone $C$ (with $x \leqslant y \Leftrightarrow y - x \in C$). A bit less straightforward is the fact that an inverse semigroup is a partially ordered set with $x \leqslant y \Leftrightarrow \exists e, e^2 = e, x = ye$.
Would you have other such examples?
Thank you.
Edit: I am not looking for examples of partially ordered sets, neither for examples where a partial order can be obviously defined; the example of inverse semigroups gives a good idea to the kind of answers I would love to have: once one gets the definition of an inverse semigroup this is not at all obvious that a partial order compatible with the algebraic structure can be defined and lead to fruitful discoveries.
Well, there are those things called power sets (or more general arbitrary sets of sets). The subset relation defines a very canonical partial order.
If you have already a partially ordered set $P$ and an arbitrary set $X$ then the set of functions $X\to P$ can be equipped with a partial order by $$f\le g \text{ iff } \forall x\in X \:\:f(x)\le g(x).$$