The modern day dry lube typically use PTFE, teflon and other agents suspended in an evaporating solvent. When the lube is applied it is wet, but after a few minutes it dries leaving a waxy protective coating on your chain. These lubes hold up fairly well in wet conditions as well, but you may need to reapply the lube more often in these conditions. If you're riding in wet and muddy conditions then a wet lube will serve you better. Wet lubes simply resist the abuse of mud much better than dry lubes.
Wet lubes are the more traditional of the chain lubes.
They go on wet and they stay wet. Wet lubes provide excellent lubrication and chain protection in clean conditions and also protect against moisture extremely well. However, where wet lubes begin to fail is in dry dusty conditions. The lube acts as a magnet to dust and can quickly clog your drivetrain with thick oily mud. The most basic wet lube is oil.