Intuition of partial-ordered set. Recently I've come in front of partial ordered set. I've read the wiki article but couldn't comprehend it especially the concept of set together with a binary relation. Can anyone please explain me what partial ordered set means?

A partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

What is the meaning of this statement? Please help.
 A: First, think about the natural numbers, $\mathbb N = \{1,2,3,\dots\}.$ I'm sure you know that given any two natural numbers $n,m,$ either $n<m,\,n=m,\,m<n$. Someone may want to generalise the notion of $<.$
We define $<$ as a relation on $\mathbb N,$ that is we define $R=\{(a,b)\in\mathbb N^2|a<b\}$ and say that $n<m$ iff $(n,m)\in R$. This definition feels a little tauntological as I have used $<$ to define $<$ as a relation.
Consider now the set $X=\mathbb N\times\mathbb N$. we might want to define some kind of "less than" relation on this set so for example we might say that $(x,y)<(x',y')$ iff $x<x'$ or $x=x', y<y'$ (here we're implicitly defining a set like $R$). After some careful thought, you should see that this is like alphabetical ordering in a dictionary.
We say that a set is ordered if there is some relation $<$ on the set that behaves like we expect so either $x<y, x=y, y<x$ and if $x<y, y<z$ then $x<z$.
The problem is that we can't always do this so there is a less strict definition:
A set is partially ordered if there is some relation $<$ on the set such that
either $x<y, y<x, x=y,$ or $x$ and $y$ are unrelated. However, we still have that $x<y,y<z\Rightarrow x<z.$
An example of a partial ordering on $X$ would be that $(x,y)<(x',y')$ iff $x^2+y^2<x'^2+y'^2$ Note that this means that the relation is not defined (for example) between $(1,0)$ and $(0,1).$
