Corsets have an amazing breadth of variety, though I often simplify it for neophytes to the two basic types, overbust and underbust, at least as a starting point. When corset shopping, the range of options and, importantly, the nomenclature around them can be quite daunting. The following is a simplified and streamlined guide to corset styles for modern corset wearers. Those interested in learning about historical corsetry in greater detail are encouraged to check out some of the wonderful books about corsetry (I particularly recommend Norah Waugh's Corsets and Crinolines), as I am not personally a fashion historian.
Overbusts
Stays/Bodice - stops around the natural waist, has straps, may have tabs, as shown above..
Midbust/Flat Front - the modern simplification of a "Victorian" corset, this is a very cleavage-friendly style with a straight-across neckline.
Edwardian/S-Curve - characterized by swooping seam lines, a flat front and outthrust derriere (the "S-Curve"), and a longline hip.
Sweetheart/Contoured bust - the modern standard, the bust is supported and rounded with a defined underbust and, of course, sweetheart neckline.
Cupped Corset - Similar to a sweetheart, but the cups are actually seamed in at the underbust and more fitted, and may have any level of coverage available in a bra (demi, full, etc). Cupped corsets, ideally, should be custom-fit with a mockup.
Underbusts
Classic Underbust - covers from the underbust (bra band level) to the lap
Longline/Edwardian - hip shaping that is low at front and side hip but scoops up over the lap. Edwardian corsets were often underbusts or low midbusts, though modern longline styles are popular with waist trainers and plus-sizes bfor their hip shaping.
Pointed - similar to a classic underbust but shorter at the side hip, with points at the top and bottom. The top line roughly echoes the line of an underwire, coming up between the breasts about an inch or so.
Ribbon Corset - a pointed underbust whose shaping is achieved by the careful laying of ribbons horizontally around the body, seamed into two to four vertical panels.
Cincher - a shorter underbust, covering the lower rib cage with a short hip.
Waspie - even shorter than a cincher, if only by a couple inches, the corseting equivalent of a wide belt. Great for styling with outerwear but more prone to creating a fold of skin at the back. Cinchers and waspies can often be worn by full-busted women as standard underbusts.
As you can see, even with just the basic styles there are many corset options available. Each designer is going to have their own interpretations and completely new styles, so you can be sure to find a corset that suits both your silhouette and intended purpose.