Broadly speaking, vermouth can be broken down into two styles: Italian, also known as sweet, red, or rosso; and French, also known as white or dry. Other designations are possible, though; some manufacturers now have rosé vermouths on the market now. Quinquinas and americanos can be red, white, or rosy, depending on the bottling.
Keep in mind, the words French and Italian here are used loosely to describe styles of vermouth, not their country of origin. Italian brands make French-style vermouths in Italy, and at least one French brand (Noilly Prat) makes Italian-style vermouths in France.
If you're not confused yet, I'm not trying hard enough; but to make things simpler, I'll refer to the styles from this point on by their color.