Why are partial orderings important? I was reviewing my old Discrete Mathematics notes, and I came across a section describing how Partial Orderings are identified. I understand this, but I can't seem to recall/find information on why partial orderings are important.
What is the significance of knowing whether a set is partially ordered?

A partial order is an abstraction of the concept of 'order' everyone is familiar with, like the one on the integers, the reals, etc. Altough this is a bit misleading because those orders are in fact 'total': any two elements can be compared. 
But in fact lots of 'orders', such as total orders, are posets satisfying additional axioms. Another example is a well-order, the habitat of transfinite induction, which is an extension of induction on $\mathbb{N}$. 
A poset in which any two elements have an upper bound is called a directed set. A function from a directed set is called a net, and this is a generalization of a sequence. Nets are quite useful in topology/functional analysis.
Every poset is a category, if we take as objects the elements of the poset, and if we take exactly one arrow from $a$ to $b$ if and only if $a\leq b$. Composition is defined beause of transitivity, and the existence of an identity arrow corresponds to reflexivity. Many basic concepts in category theory can be illuminated by this special choice of category. But of course this is only useful if you already know/care about posets in the first case.
