A dry clutch is just that, a clutch that lives and works in a dry environment. Usually, but not always a single disc between two pressure plates or one pressure plate and the flywheel. Usually, but not always mechanically controlled, a dry clutch can sometimes be hydraulically controlled. Dry clutches are cheaper to build, cheaper to replace, and generally do not last as long as wet clutches.
A wet clutch is a clutch or clutch pack that lives and works in an oil bath. Usually, but not always a stack of alternating lined and unlined discs with alternating internal and external splines within a drum and having a shaft extending through the assembly. The shaft and drum are the drive and driven elements, one way or the other. Usually, but not always hydraulically controlled by an oil filled piston that compresses the discs directly, or a hydraulic piston that controls pressure plate fingers. Wet clutches are more expensive to build considering the control circuits and clutch together and usually more expensive to repair or replace than dry clutches. Wet clutches will outlast dry clutches in like operating conditions by a wide, wide margin. This longevity and the ability to marry wet clutches with computer controlled modulation and engagement adds to their popularity and use.