Dry and Wet are universal terms indicative of the conditions the lubricant is meant to withstand. Most chain lubricant manufacturers offer these two variants as a minimum. As a very simple rule of thumb, use Dry lube in dry riding conditions and Wet lube in wet and muddy riding conditions.
These go onto the chain wet, but then dry to a waxy finish. Typically, Dry lube takes a couple hours to dry, so plan accordingly. A positive aspect of dry lube is that it tends to not collect a lot of dirt, and hence this makes it perfect for cycling in dry conditions. On the downside, dry lubes wash off very easily and require re-application - especially if ridden in wet conditions.
Wet lubes are thicker and stick to the chain to form a protective layer that remains wet to the touch until rubbed away. They are ideal for wet riding conditions, as they offer a highly increased resistance to rain and are harder to wash away. The downside is that they collect dirt and grime easily and hence the chain will require regular cleaning to wipe away the dirt and prevent it from chunking up and damaging the componentry.
Using Other ‘Non-Specific’ Lubes - A word of Caution!
A classic faux-pas when it comes to lubricants is to use a lightweight household oil such as WD40, which is designed for low use parts. Whilst this will grease the chain in the short term, it is not meant for outside use and will very quickly wash away. The other extreme is motor oil. This is generally too thick for use on a bike chain and will not penetrate the smaller parts of a bike. It is also very sticky and will pick up dirt from the road very easily.