As Shadowrun is, to paraphrase The Comic Book Guy from the simpsons, my "Favorite. Game. Ever!", I figured I should chime in here.
Shadowrun has come along way from first edition. The rules have improved alot, the setting has become a bit more cohesive, and the timeline has taken some interesting turns (The Shadowrun timeline, in terms of cannon setting, is always progressing, at realtime... IE, each year Shadowrun exists in the real world, the Sixth World (Shadowrun's setting) progresses another year, or pretty much.).
The rules... Hmm... In a way, they are like a much, much grittier version of the White Wolf Storyteller system. By which I mean, dice pools (D6's, as opposed to D10s of Storyteller), generaly low-ish scores (Physical attributes range from 1 up to the midrange of single digits for the average person, for example, before cyberwear is accounted for... Not the 8-18 of D20, for example.) Depending on how many of the systems you use strait from the book, it can be quite complex... Decking (aka hacking), for example, involves such details as making sure your deck has enough memory to run all the the programs you are running, how fast various sub-systems of the deck are, all sorts of nifty details. Likewise, if you use the advanced, custom firearm construction rules in The Canon Companion, you have all sorts of options for making a gun from the ground up... What you want the components made of, if you want custom grips, various methods of recoil surpression, stabilization, sights, targeting assist mechanisms, etc. The flip side of this is, most Shadowrun GMs eventualy will choose a few systems that they think are a bit too complex for the type of games they run and trim them down. Decking and Astral Projection are the two most commonly trimmed, but I've only trimmed Astral Projection a touch... Decking is actualy fairly easy once you get the hang of the system.
Cyberpunk Fantasy is the best description of the setting, but it doesn't really explain much...
Like someone else mentioned, most major governments, at best, are fairly fragmented. The United States, for example, exists only as The United Canadian and American States, and is probably half the size of the USA... And occupying only the north-eastern 1/4 of what used to be the USA (It has some of Canada now, remember). There aren't a whole lot of nations that have not had their borders altered some, but there are a few... Mostly the smaller ones, but again, not always. Japan is intact, though the government is different. Australia. Some of the european nations (Spain, Portugal, France, Norway, England, Ireland (Renamed Tir Na Nog)...), a few others. Many, many new nations have sprung up... Some are racial (Tir Tairngire is a nation occupying more or less the same geographic are as the state of Oregon in the USA, run, to paraphrase a common saying, "By Elves, For Elves".), some are political, some are other things.
Japan is now the economic focus of the world in some respects... The closest thing to a world standard currency is the NuYen, and Renraku is one of the most famous megacorps.
The specifics of the setting are vast and I'm not going to go into too much detail, but some quick common themes: Magic and Technology aren't exactly opposites, per say, but they don't get along well. Modification of the body via cyberwear, for example, makes the body "less whole", and damages a persons ability to use magic. There are dragons. The world, in many places, has become very much a throw-away culture... There are vending machines for almost anything you could want, clothes, electronics, you name it. There are traditional stores too, of course, but cheap clothing can be gotten from a vending machine if you need to change your outfit quickly. Space in the major cities is at a premium... The capsule hotels Japan is known for are much more common everwhere now, housing tends to be smaller, more packed into any one building than used to be, etc. If you live a legit life, you are tracked in pretty much everything you do... Not quite to a "Big Brother is Watching you 24/7" level, but if the government (Or other interested parties) want to find out something about you, they probably can... If you payed for it with a legit credstick, or purchased a plane ticket with a valid SIN, etc.
Some other supplemental information:
http://www.divnull.com/lward/sixthworld/
Good maps of the world as of 2062. I'd suggest you download the PDF map (Click on Acrobat World Maps, on the left)... The link to the dev-null site isn't broken per say, but I've never had it work by clicking on it. Instead, just copy-paste this into your browser bar (
ftp://ftp.divnull.com/pub/srun/maps/), and download as many of them as you want... Get "World2062.pdf" at least, though.
http://matrix.dumpshock.com/mcmackie/
Go there to download the best shadowrun character generator software(Actually, the only one I know of anymore... Used to be a couple others, havn't seen them in a while), will give you a peek at the system. And if you plan to play, it shortens the time needed to make a character considerably.
Someone already posted a link to the dumpshock forums, so I won't bother with that...