Yesterday I reinstalled my drive shaft. The carrier shaft center bearing had failed, and sidelined the Ford until I got it replaced. This is a non-trivial job that often requires the loving touch of a driveline shop; on some vehicles the bearing can be trivially replaced, but on most (including this one) it must be pressed on and off.
On my way up Cobb Mountain I started hearing and feeling click/thump noises from somewhere in the middle of the truck. It turned out that the carrier shaft bearing's rubber isolator had failed, probably because of added vibration because a bolt was falling out. I was lucky enough to have an appropriate wrench handy, and made it home; as long as you are very gentle on the accelerator you can even go up hills with the isolator broken, but jam that pedal and THUMP THUMP THUMP as the bearing bounces around in the bracket.
The word transmission refers to a transfer of something. In this case, that is kinetic energy. In basically every automobile, a rotational force (whether produced from a reciprocating motion, such as in a typical gasoline engine with pistons, or from a rotary engine, electric motor, or turbine engine) is both transmitted from the engine to one or more driven wheels, but it is also geared down (and sometimes also up) so that it can do useful work.
The transfer case is the part of a four wheel drive vehicle which distributes power between the front and rear axles. All wheel drive vehicles have a differential instead. The transfer case differs primarily in that the front and rear wheels are driven at the same speed. Typically, the transfer case also allows the disengagement of the drive shaft leading to the front axle, and it usually also has a neutral mode which allows disconnecting both axles for the purpose of towing the vehicle.