You can see the natural examples of partial orders in the Wikipedia page. In mathematics, though, once an object is well-defined and has some nice properties, it automatically becomes important regardless of applications or how arcane it may seem to some.

Example: all natural numbers are interesting. Indeed, suppose the contrary, that there are some boring numbers. Then there is the least boring number by the minimum principle (which is the contrapositive of strong induction). But hey, this is interesting!

That said, this forum is not for questions that require student's own work to earn credit (see rule #6 here).