Specifically what types of beer would you use wet hopping for and how does it affect the flavor/process differently than dry hopping a beer?
Wet hopping is adding fresh hops to the fermentor to impart flavor and aroma to the finished beer. The principles are the same as for dry hopping - the hops are added to secondary (or to primary after fermentation is complete) 3-5 days before racking. The main difference is the hops are fresh, i.e. "wet" - the freshness gives a more vibrant aroma and cleaner flavour.
Fresh hops weigh 4-6 times more than their dry counterpart, so this needs to be taken into account when weighing. For example, if you normally dry hop with 1/2oz, then you'll need to wet hop with around 2-3oz of fresh hops.
Introducing spoilage organisms into the beer from fresh hops is unlikely, especially when added after (primary) fermentation is complete. Even so, you may want to put the hops in the freezer overnight to kill any pests, although this makes them slightly less fresh.
With fresh hops, the most desirable qualities that distinguish them from dry hops are the freshness of the volatile oils. This makes them best suited to flavoring and aroma additions, which also means there is less need to worry about the IBUs, since they contribute relatively little bitterness compared to the bittering additions. When using fresh hops in the kettle, remember that they contain a lot of water, which will be added to your wort. If you want to hit your SG on the mark, you'll need to compensate for the water added. Assuming wet hops weigh 5 times more than dry, then 4/5s of that weight is water. For example, 2oz of dry hops is 10oz when wet. Since 2oz of the wet hops is hop matter, the remaning 8oz is water.
Hops are normally picked from the bines (yes with a 'b') and quickly brought to a processing facility where they are dried in a controlled environment. The hops are then vacuum packed and set out whole or further shipped out for pelletization. These are all dried hops.
The term wet hopping is used to describe hops that are picked, and in the same short order, brought to the brew house and used. Generally that happens same day as picking or the next day. Wet (fresh) hops are very perishable...which is why they usually get dried for storage reasons.
Dry hopping is a technique term that normally describes the process of adding hops (wet or dried) to finished beer. Many breweries that have access to "wet" hops will indeed dry hop the finished beer as well as use them in the boil.
There aren't any limits to using wet hops if you have a supply of fresh/wet hops put them in any beer you like and see what the results are. The types of beer that use wet hops are typically are pale ales and IPAs. Any style that is hop forward and where you want hop character to shine.