I'll give it a shot.
FIM (initials for International Motorcycle Foundation, in French) is the Geneva-based international sanctioning body for international motorcycling events. Most national organizations (eg AMA) are affiliates of FIM. FIM is analagous to FIA (for Autos) and FIS (for Skiing) -- probably no accident that these organizations are also based in Geneva.
Rules etc for GP (125, 150 and MotorGP), World Superbike, World Supersport, World Sidecars, World Endurance etc are issued by FIM and can be found on their website.
GP racing is actually run by Dorna, a private company and World Superbike, World Supersport, and I think Sidecars, are run by SBK, a unit of publically traded Octagon Ltd (although Octagon's racing activities -- they also own a bunch of British race tracks -- are for sale). I am not sure who is the operater of Indurance racing, or MX, Trials etc.
I believe that Dorna, SBK etc have a great deal of influence over just what the rules are, and that, to a large extent, the FIM "rubber stamps" them, but I don't have first hand knowledge of that.
AMA (American Motorcycle Association) is a non-profit membership organization, and is the blanket organization involved with all sorts of motorcycling activities in the US -- most of which have no involvement with racing at all. Their racing activities are handled by a for-profit subsidiary, AMA Pro Racing. As I mentioned, AMA is loosely affiliated with FIM, which is how they arrange for the joint AMA/World Supers weekend at Laguna. I believe that there is a closer association in MX where some rounds of the AMA champinship also count to the World Championship.
There are several other large national or regional road race series (eg WERA, CCS etc) that are not affiliated with the AMA.
At the local club level in the US, organizations may or may not be affiliated with the AMA. Those that are, generally still operate entirely with their own rules -- I think it is largely a matter of insurance (smaller clubs can get better rates through AMA or one of the other large groups than on their own)
Most large countries have their own national racing series -- I can't speak for their off-road racing, but at the Superbike level, there are large, professional race series in Canada, Britain, Australia, Italy, Japan and probably others. BSB, the British Superbike series is owned by Octagon (who also control World Superbike) -- I don't know much about the structure of the others.
From a fan perspective, what is probably most relevant is summarized as follows:
GP Bikes
Purpose-built pure race bikes with no production components. 125s are all 2-stroke with single cylinder. A number of makes compete, some of which you likely will not have heard of if you are American (eg Derbi, Gilera etc). Aprilia tend to be very strong in this class.
250gp are all 2-stroke twins. Aprilia also are most dominant, although in 2001, the late Daijiro Kato won a record number of wins on the way to taking the championship on a Honda. Yamaha are also sometimes competitive.
MotoGP are mostly 990cc 4-strokes, although 500 2-strokes are allowed -- only Proton, for the first few races of this year continue with the 2-strokes. Prior to 2002, the class was limited to 500cc and for many years, had been all 2-strokes. Honda has historically dominated in recent years (Suzuki's 2001 championship by Kenny Roberts Jr being the notable exception). This year there are more viable competitors, however, with Yamaha, Ducati and Aprilia already competitive. Suzuki still lags a bit and Kawasaki is still far from competitive.
Superbikes
World Superbikes are production-based machines, with fairly extensive modifications allowed. The fact that factory bikes cost well over $500,000 each gives a pretty good idea of how extensively they are modified! Historically, they allowed 750cc 4-cylinder, 900cc 3-cylinder and 1000cc twins. In recent years, the twins have dominated, with Ducati and Honda trading wins and championships. For 2003, the rules were changed to allow 1000 cc 4s, but with restrictors in the inlet. Because of delays in issueing the rules, as well as economic issues, all the factory teams other than Ducati and Suzuki pulled out. There are a number of privateer teams however, most with Ducatis (last year or earlier models) and a few Yamahas and Hondas. In the first two rounds (actually 4 rounds as they run 2 races per weekend) the Ducatis continue to win, but the Suzuki has had huge improvement, and I expect they will get some wins before the season is over.
World SuperSport is for 600cc production bikes (up to 750cc for twins) with limited modifications. They run DOT (or Euro equivalent) tires on stock wheels. I am not sure of exact mods, but basically exhaust, shocks etc.
AMA
AMA runs mostly production-based (except for the 250 gp class that is in its final year). In general, rules are broadly similar to World superbike.
Superbike -- unlike World Superbike, they allow the old 750s to be bored out to 800cc. In addition, they allow 1000cc 4s with no restrictors, but with more limited internal modifications (eg stock crank, rods, valve sizes etc).
SuperStock -- this year, is for production based 750cc bikes, with similar rules to World SuperSport above. Next year this is opened up to 1000cc bikes. Fairly close to stock, except that the factory bikes are mysteriously much faster than the rest. Basically all GSXR 750s, with a few Kawasaki 636s and some 600s running up one class.
SuperSport -- almost identical with World SuperSport above, except that 750 twins are not allowed. The most competitive class and not changed for next year.
Formula Xtreme -- this year is largely overlapped with Superbike,which is why next year is totally different. Limited to production-baded bikes (I think they must be US legal, where Superbike requires only that they be legal someplace in the World). I think that the mods are more generous than Superbike, but most Formula Xtreme bikes now are also Superbike legal, and riders do sometimes cross over.