Wet leasing indeed means the plane comes with the whole crew, fuel, catering (usu.), everything from soup to nuts. It's basically some other operator's flight flown on behalf of the airline that sold the tickets but didn't have the capacity to operate the flight itself. This is an extremely expensive way to lease an airplane, and it's typically charged either by flight hours or block hours (the time from when the blocks are removed from the tires at the origin to when they're put in place at the destination, which can be a big difference from flight time). The lessee airline doesn't have to worry about maintenance, crew, or anything except putting pax in the seats.

Airlines resort to wet leasing most often to cover a peak season demand or some other demand spike or sudden capacity loss (aircraft in fleet unexpectedly not airworthy). Sometimes the lessor slaps the lessee's decals on (which results in some funky hybrid liveries), sometimes they don't bother. An airline might wet lease an aircraft under circumstances when it's not over a barrel to meet demand. That might occur when two or more airlines have different peak seasons, and one wet-leases to the other when the first is on a slow season and the second is in peak. Those arrangements are rare and difficult to coordinate. Another might occur when one airline has a plane stuck in a remote location for twelve hours before its return-to-home flight, so it wet-leases the plane to a local operator for shuttle flights to keep the a/c in service-- also extremely rare. Wet lessors are operators, not strict lessors, though they may be an operating arm of a leasing company.

Dry leasing is a more traditional lease, kind of like leasing a car but aircraft-to-car analogies are very clumsy and best avoided. The leases are incredibly elaborate, with all kinds of delivery and redelivery conditions, montly rent, maintenance reserves, security deposits, operations requirements, and on and on and on. The lessee might have a purchase option, but basically it has to give the plane back to the lessor at the end of the term unless they agree to extend. The major lessors work with these a lot.

Lease-financing is more like rent-to-own, just a more strictly controlled way of financing a purchase than a straight "mortgage." Companies that don't necessarily have a lot of technical types deal with lease-financing more than dry leasing. Those would include companies that treat aircraft leasing as a side business.