Priming an engine is the act of preparing the engine to fire. Most engines have some form of dedicated primer, but some do not. Priming systems in horizontally opposed flat recips generally deliver fuel directly to the cylinders, placing a charge of raw, unburned fuel directly in the combustion chamber.
Only a very small amount of fuel is necessary, and directly priming an engine can cause an afterfire, which comes out the exhaust stack. One should prime only the bare minimum necessary (this doesn't apply to radial engines, which generally take a lot of prime through the supercharger when cold, and will probably catch fire anyway).
Pumping the throttle is using the accelerator pump, and this only works on engines equipped with an accelerator pump. Many airplanes don't have one, and it's quite comical to watch a pilot pumping the throttle out of habit when he or she doesn't understnad the system, or know that no accelerator pump is on the carburetor.
Watching someone pump the throttle on a fuel injected engine is a lot of fun. Usually when asked, the response is, "my instructor taught me to do it." Apparently this excuses the lack of ability to think for one's self, but I digress.
Use of the accelerator pump places a very small squirt of fuel into the induction manifold. In most light airplanes, the carburetor is an updraft arrangement, and the fuel squirts up...and runs back down through the carburetor and into the carb air box. It collects there, ready for a backfire, which is the proper term for an induction fire (a fire in the exhaust being an "after-fire). The carb air box has a drain hole, which permits the fuel to run into the cowling, where it cannot be contained once ignited...until it's burned itself out.
If one intends to use the accelerator pump during start, one should pump during the start However, this places the throttle plate in a position to be opening fully and closing fully repeatedly, while raw fuel is sprayed into the induction. The liklihood or potential for a backfire increases dramatically as the mixture changes back and forth; the potential exits to set fire to the carburetor and air box, as well as do damage to the induction by a backfire.
If one is to use the accelerator pump at all, it should be a couple of quick pumps before starting, and perhaps one as the engine turns. The throttle should remain closed during the start itself. The accelerator pump is not a priming tool; it's purpose is to enrich the mixture briefly during a power increase to make a smooth power transition. Nothing more. It's there as part of the automatic function of the carburetor, not as a tool to get the airplane started. Let the idle jet do it's job.
One may also prime some fuel injected airplanes with an auxilliary fuel pump (boost pump). This will work on some airplanes, but not on others. Again, know the system. On some airplanes, pushing in the mixture and applying boost will do nothing more than apply pressurized fuel to the controller. Until the engine begins turning, nothing more happens, and openign the throttle does nothing more than open the air valve butterfly, or the throttle plate. On others, one can put fuel directly to the cylinders via the fuel injection nozzles, but can easily flood the engine.
One needs also to pay attention to the damage done during priming and the subsequent start. Fuel is a solvent, and once placed in the cylinder in liquid form, it quickly removes the protective lubricant from the cylinder walls during start. The result is an increase in cylinder wear by an order of magnitude. (or two). Don't overprime, and don't prime when not necessary.
Radials don't see the common use they once did, but you may occasionaly see one if doing a checkout in something like a Cessna 195. Overpriming a radial engine, aside from fire issues, has the very real hazard of hydraulic lock in the bottom cylinder. Liquid lock, or hydraulic lock, is usually the result of trapped oil in the lower cylinders. However, the same thing can occur during start by overpriming some engines, and in the event of a valve malfunction, or a fuel flow high enough to preclude being drained through the valves fast enough, the result can be a bent slave connecting rod (and subsequent engine failure).
In the winter, a radial often needs a lot of prime, through the supercharger, to get a good start. The liklihood of a fire goes up exponentially, and one must be aware and prepared. For most radial supercharged engines, the fuel dumps directly into the supercharger from the carb, or from an aux boost system, and doesn't go right to the cylinders. Accordingly, the cylinders still aren't getting a raw fuel mixture. It's not uncommon, however, to fill the supercharger 1/3 full of avgas during the start when it's really cold...leading to the issue of the supercharger drain creating a massive fire hazard if you elect to discontinue the start. (The C-97 was famous for induction fires as the fuel would pool in the inside of the airscoop, and the drain hole would invariably be plugged with congealed oil).
Prime where necessary, but only where necessary. If you are in the habit of pulling the propeller through in the winter (not a good idea), priming first will only dramatically increase cylinder wear and reduce any available residual lubrication on the cylinder walls and valve stems. I see a lot of people prime, then pull the prop through; it accomplishes nothing but damage.
One final controversial note: after-shutdown priming. When operating a fuel injected engine to remote locations, I made a habit of opening the oil filler and check doors on airplanes such as the Cessna 207, to act as an additional vent to cool the engine compartment. In order to facilitate start-up again, I often flooded the injector lines using the boost pump after shutdown. There are those that will debate the safety and efficacy of this technique, but I still practice it.
If you do it, however, be aware that you've just placed an unburned charge of fuel in the cylinder after shutdown. There are obvious safety implications. In the back country, I felt I could mitigate these implications and gaurd the airplane without fear of someone inadvertantly moving the prop or causing the engine to fire (it can fire on it's own without outside intervention, so bear that in mind). I felt and do feel that given the alternative of being stuck in the event of a hard start, anything I could do to assist this was to my benifit. It's not something I practice with a student (on the rare occasion I get a chance to instruct, any more), but it's still a viable practice under the right circumstances. Let the flames begin.