Stars are plotted according to their temperature (horizontal axis) and luminosity (vertical axis). Because temperature and color are correlated, you will sometimes see the horizontal axis represented as a color axis instead of a temperature axis. This comes as no surprise. We know from everyday life that color and temperature are correlated; if you heat a coat hanger on a stove top it will first glow red. Then as it gets hotter, orange, yellow, and if you can get it hot enough, blue and white. Same with stars.
Another popular horizontal axis that you'll often see has to do with a star's spectral classification: O,B,A,F,G,K,M (Oh, be a fine girl, kiss me). Type O stars are the hottest, type M the coolest. The Sun is a type G star.
Likewise with the vertical axis. Luminosity and absolute magnitude are directly related. So you'll often see the vertical axis represented as absolute magnitude instead of luminosity.
When you plot all the stars within 25 parsecs of the Sun, a surprising pattern emerges. First, there is the swervy-curvy line that stretches from the upper left to the lower right. It turns out that this is where stable, mid-life stars, called "main sequence" stars, reside.
As stars like the Sun move into old age, they nova, blow off their outer layers and migrate off the main sequence into the upper-right region of the H-R diagram. These are the red giants. What remains of the star, the part that didn't blow off into space, collapses down to a dense hot object called a "white dwarf," or perhaps, if the original star was massive enough, to a neutron star.
There is so much more. I'd encourage you to Google H-R diagrams and to do a Google image search for them as well, and check out all the different forms. You can tell a lot about a star from it's position on an H-R diagram.
Well, I'm about chatterboxed out! And you are as well, I imagine. Ask a chatterbox for a few sentences and you'll get paragraphs! :-)
I did not know there is no helium ash build-up at a red dwarf's core, thanks!