We have defined: a Morse function on $X$ is a smooth function $f:X\rightarrow\mathbb R$ with only non-degenerate critical values. I tried to get some intuition about this, and found the section Basic concepts of the Wikipedia-page on Morse Theory to be very useful (http://en.wikipedia.org/wiki/Morse_theory >> Basic concepts).
Can someone explain to me how Morse functions fit into this intuitive explanation? In the section afterwards, Morse functions are neatly defined, but not anymore intuitively explained.