If $a$ and $b$ are two numbers on the real line, we compare $a$ and $b$ by knowing which of them comes first as we move from $-\infty$ to $\infty$ on the real line.
However when $A$ and $B$ are matrices, the comparison is through definiteness. We say $A \succ B$ iff $A-B$ is positive definite. Positive definiteness of $A$ means $x^TAx>0\ \forall x$; essentially the function $f(x)=x^TAx$ takes the form of a bowl with its base at origin.
How does this "bowl" help in comparing two matrices? What is the intuition behind using definiteness in matrices for ordering?