Ribbon mics are a type of dynamic mic that uses a thin ribbon as the element instead of a diaphragm. They were some of the first quality mics ever made and have become very popular and trendy in recent years for a good reason. They sound great. They are very different from condensers and even other dynamics. They have a slower transient response than condensers and other dynamics. Why, you might ask, would you want a mic that is LESS accurate? Well, especially in the age of digital recording, perfectly accurate recordings with great transient response can sound kind of harsh and brittle. Back in the day when all your favorite rock and jazz records were recorded, they were recorded to tape (Something we can, and still, do here at The Hangar). Tape also does not reproduce transients perfectly and tends to soften and limit them. People like the way this sounds. People like the sound of tape. People like the sound of Ribbon mics when they record onto digital systems. Recently ribbons have become popular again as drum overheads.
Keeping the above in mind, other popular uses of Ribbon mics are on reed instruments like harmonica and saxophone as well as strings. Steve Albini popularized their use on guitar amps. Lot's of engineers like them as vocal mics. In all instances, the above uses tone down transient information like scratchy strings for instance and results in a warmer, richer sound. You might prefer a ribbon mic for a string section for instance while you might want a condenser for a solo violin line. George Massenburg has been quoted saying that he'll use Ribbons on vocals for the softer top end but then boost the real high end above 12k or so with a good EQ for a really natural, 'airy' sound.
PHANTOM POWER WILL DESTROY RIBBON MICS!!!! It is super important to make sure you have the phantom power off before connecting a ribbon mic.