In the six years since Damodred did the above writeup, much has changed in EVE, but at the same time very little has changed. The basic mechanics of game play are still as described, the sounds have improved (although a stock catchphrase in Goonwaffe is "Eve has sound?") and the various visual effects have been improved, to say nothing of the art used for ships and planets. This has required upgrades to the EVE client, but you can still run the client on most any off-the-shelf desktop PC. There has also been a minor change to training, which allows new players to progress faster through their basic skills, but prevents veteran players from taking vacations from the game while characters train long (2-month) skills. Now, if your character isn't active (paid up), it stops training. On the other hand, it is also now possible to buy "PLEX licenses" in the game for the universal currency of isk, so a moderately competent player can actually play the game for free by generating about 250 million isku/month, depending on the market price of PLEX licenses.
The other major area where things haven't changed is the basic "map" of the game. The core of the game is made up of the high-security star systems belonging to the Amarr Empire, Caldari State, Gallente Federation, and Minmatar Republic.* Players choose which of these their character will start in, which will determine what their initial skill set is and what type of ship they'll fly. Inside "highsec", random aggression/griefing against other players is discouraged by the NPC police force known as CONCORD, which will spawn a number of ships to destroy an aggressor's ship within seconds of the offender opening fire. It is also indirectly discouraged by a security status for each character, which cannot be hidden, and a bounty system that allows players to hunt down other players without fear of CONCORD intervention.
In and around highsec systems like fat in well-marbled beef are lowsec systems. Here, the rewards from mining and ratting (shooting NPC pirates in the asteroid belts) are higher, but so is the risk of being attacked by other players, who unlike the NPC pirates may destroy not only your ship, but the pilot pod within it. Being podded doesn't end your game play; your clone wakes up in whatever station you began at (or another you've moved it to) without whatever brain implants you might have had. Also, since clones only retain a limited amount of skill points, if you've failed to upgrade your clone to keep pace with your increased skills, some of them may disappear.
The vast majority of EVE players spend all their time in highsec and occasionally in lowsec, raking in isk through increasingly more rewarding missions, building vast piles of wealth through production and trade, or taking part in faction warfare, wherein they join militia forces of one state or another, and fight their enemies for control of contested systems on the borders. However, most of the news of EVE is generated in nullsec, where mighty player alliances claim space and fight battles involving hundreds of combat starships on either side. Unlike Empire space, where a player can stay in his original NPC corporation forever and do quite nicely, in nullsec you have to have friends, preferably friends who speak your language and who can be relied on not to sell you out to another corp or alliance. You and your friends form a corporation, go forth into nullsec, and most likely either join an existing alliance to which you have some tie (usually language or nationality) or make an agreement with an alliance to rent space from them. It's possible to operate in NPC nullsec space without doing these things, but since most nullsec alliances operate under the rule NBSI - Not Blue (Friendly) Shoot It - your time in nullsec is liable to be brief and exciting, but not in a good way.
So why bother? Why not stay in Empire, or in lowsec, where the risk is a lot lower? Two reasons. The payoff for playing in nullsec, where the richest ores and most lucrative rats are, is orders of magnitude higher than it is in lowsec. You can easily kill enough rats in a month to play EVE for free after you cash in the rat bounties and sell what you looted and salvaged from the wrecks. You can make money producing ships, modules and ammunition for your corp mates who are out shooting rats or enemy corp members, at a much higher profit margin than you ever could in Empire. Quite aside from the payoff, though, is the metagame: the game of empires that keeps nullsec in a constant state of flux as alliances break up and reform, wage war on each other or merely fight guerrilla wars against those corps who rent from hostile alliances. I personally think that's what attracts most players to nullsec - they want to be something bigger, to be part of making history in a game whose designers have for the most part left that history for the players to write.
*It's worth describing the basic attributes of these four nations/races in terms of their ships and weapons, because almost nobody who plays the game cares about the background history of these nations except the handful of role-players.
Amarr ships mainly use lasers, are heavily armored, and are vulnerable to EM and thermal damage; the lasers mainly do EM and thermal damage. Beam lasers can hit at longer ranges but do less damage, while pulse lasers have shorter range but do more damage. Amarr EW uses the tracking disruptor, which breaks the lock of turreted weapons. This isn't too useful, since their main opponents (the Gallente and Minmatar) are fond of drones and missiles, which don't have tracking issues and are unaffected by tracking disruptors.
Caldari ships carry missiles and railguns (mostly the former), which allows them to deal out not only EM and thermal damage but kinetic or explosive depending on which type of missile they have loaded. Railguns mostly do kinetic and thermal damage and can hit targets at long range. Caldari ships rely on shields for defense, and the Caldari are also very good at ECM, which can jam enemy ships' sensors and make them unable to get a lock on targets.
Gallente ships carry lots of drones and blasters, which are to railguns what shotguns are to rifles. Gallente ships rely for defense on armor, and avoid being sniped to death by using remote sensor dampers, which make enemy sensors less effective and force ships to come to close range before they can get a target lock. Gallente ships are also ungodly slow and clumsy, and are currently considered the worst ships in the game.
Minmatar ships look as if they'd been hurriedly assembled from stolen televisions and bicycles, but in fact are some of the best ships in the game, and the most difficult to train for. This is due to the fact that most Minmatar ships use both projectile guns and missiles while relying on speed, shields and armor for defense. Projectile guns come in two flavors: artillery, which fires slower but does more damage at long range; and autocannons, which have a higher rate of fire but much shorter range. Both can fire a wide variety of ammunition that does all four types of damage in different combinations while having a bewildering variety of effects on your guns' range. Minmatar EW focuses on the target painter, a sort of laser pointer that makes its target appear larger and therefore easier to hit.