The most common partial order is the ordering by inclusion of subsets of a set $X$. The subsets are only partially ordered, since it is possible in general to have two subsets $A$ and $B$ of $X$ for which you neither have $A\subseteq B$ nor $B\subseteq A$.
This occurs even when you restrict yourself to special kinds of subsets: the collection of all subspaces of a vector space; the collection of all subgroups of a group; the collection of all subextensions of a field extension; the collection of all subvarieties of an algebraic variety; the collection of all linearly independent subsets of a vector space; etc.
One important situation is when one has a task to complete, and there are several ways to "get started", or several ways of "partially completing" the task. For example, you may be in a vector space and you want to find a basis. You can start by taking any nonzero vector, then pick a different vector not in the span, then a different one, etc. This gives you a "maze" of paths towards completing a full choice of basis. You can think of this maze as a partially ordered set (by ordering the linearly independent subsets by inclusion). A very powerful and useful result about (some) partially ordered sets is Zorn's Lemma, that tells you that under certain circumstances, you can be assured that there is at least one way of "completing the task", even if it would take infinitely many steps and there are many choices at each step.
Generally speaking, partially ordered sets are ubiquitous, so the more you know about them the better. Much like positive integers: they show up all over the place, and often you want to do things with them, so you better know what they are and what you can do.