A buzzcut uses electric clippers to shave the hair to a consistently short length that's almost-but-not-quite bald.

A quiff's most-prominent feature is its little mountain of hair at the front, combed upwards. Behind that, hair has been combed forward

A pompadour is very similar to a quiff, but features the hair slicked upwards and back, with a pomp at the front. Think Elvis Presley's hair.

First off, a fade is when the hair on the sides of the head gradually gets shorter the farther down it goes. A low fade is when the hair disappears more than a few inches below the hairline.

In comparison to a low fade, a high fade is when the hair completely disappears within a few inches of the hairline.

....and a "mid fade" is a term you can use if you want the fade to end somewhere in between a low fade and a high fade.

A high top fade is a special kind of fade—it incorporates a fade on each side with long-but-flat hair on the top.

Like the buzzcut, a crew cut requires electric clippers to shave the hair shorter on the sides, but the hair is left longer on top.

A side part cut is when a relatively short cut is clearly parted. There's some variety regarding length on a side part that you can discuss with your barber.