"Dry" lubricants are lighter weight lubes, often containing an evaporative carrier with teflon. Apply it lightly, run it in on the stand to penetrate the rollers, wipe it clean and you get a chain that doesn't have a gloopy coating of stuff on it to pick up a lot of dirt, dust, and grit (although, as ItsJustMe pointed out, if you ride in really dirt/dust prone conditions then almost nothing will solve it totally.)

"Wet" lubricants are high viscosity and coat everything with a protective barrier of gloop. Great for super muddy MTB and CX courses, commuting in the rain or snow (unless it's super-duper-cold) or any conditions where your chain might see prolonged exposure to water. The downside is that wet lubes suspend particles of everything, eventually turning a wet lubricant into a grinding paste if you don't thoroughly clean and relube frequently.