Don't Fear the Lab Mode

One way to understand how it works is to click on a swatch at the bottom of Photoshop's toolbox to bring up the Color Picker dialog box. Notice you have HSB radio buttons on left and Lab on right. Select B. You'll see a rainbow of hues at the top of the square field on left declining into gray at the bottom. You also have a vertical gradient with slider triangles. Move the sliders all the way to the top to see the most vivid hues declining to white at bottom. (If this were HSL as opposed to HSB, setting L to white would turn the whole field white; the brightly saturated hues occur at L: 50%.) Move the slider down to black and the whole field turns black. (That's the way it is for HSL, too.)

Now select the L in Lab. Move the gradient triangles to half mast -- 50% gray. There are all of your colors at full intensity around the perimeter, declining to gray in the center. (Radial projection this time.) Now move the triangles up and down. You never lose color. Amazingly, even at full Lightness and no Lightness, colors survive. That's where those imaginary colors come into play -- galactic falafel, sinister pine cone, and the rest. These are colors (with made-up names, of course) that have no chance in hell of reproducing in RGB or CMYK. (They won't disappear, they'll just clip.). It's fairly fascinating stuff.

So, to answer your question, a and b are color axes that intersect at 90 degrees. The intersection is 0% saturation, the farther you move out from the intersection, the more intense the hues get. With Hue and Sat, all colors are functions of Hue, while Saturation controls their intensity.

If this is something people are interested in, I can write it up with a few illustrations.