Who are we? Goons, and Brosefs. A Goon is simply a member of the Something Awful forums. As an alliance we are a collection of corps with similarly minded people from internet communities, with many that just became our friends through the swirl of Eve Online gameplay.
Or if you prefer, something a bit less serious.
We move in next door to our victims, into NPC stations from which we cannot be evicted. We leave broken fridges on the lawn, a car on bricks in the drive, and shoot at their pets from our porch. After a few weeks of this, other miscreants, seeing the chaos, move in as well, and before you know it the locals have broken up with their spouses, boarded up their houses and left us with a series of deliciously outraged but ultimately unenforceable cease-or-desist orders in the form of forum porn.
We just finished this process in the South-East, where we squatted like an ugly presence on the outskirts of Atlas Alliance's space, and the net result is that they have virtually none of their pets or local allies left, while Russians have started to take over their space. This is how we play Eve these days. If we shoot inanimate objects, it is purely to pick a fight.
A key advantage of an alliance the size of Goonswarm is that it can offer you a huge variety of things to do.

Simply the best Stealth Bomber pilots in Eve, they can destroy an enemy's plans before they have begun, and destroy whole fleets in the space of thirty seconds. When hostiles see Bomberwaffe in a system, their first response is often simply to try and run home.
providers of PvP to the terminally lazy. This squad sends out scouts to find targets while you sit around watching porn or reading those horrible manga comics you pretend are art. When they find one, you switch to Eve, get dropped instantly on the heads of the ambushed unfortunates, get a fight then go back to doing what you like. This terrifies the nearby locals.
Rebuilding after many of the previous incarnation of Black Ops fell out with our last CEO, these people patiently deny people the use of their own space, preying on the careless and the desperate in order to drive enemies out of their homes.
For the returning Vet who regrets training Gallente: an entire group built around using the Dominix to kill hostile capitals or defeat larger, less disciplined gangs.
We fight in other people's space. Gone are the days when fleet ops meant lining up in battleships. We now tailor ships to what we're doing, and reimburse a huge variety of Heavy Assault Ships, Recons, Heavy Interdictors, and even battlecruisers that any newbie can be in within weeks of starting the game. Before that, you'll be given frigates for free and cruisers for virtually nothing. What's more, we have Jita-priced ships, fully-fitted, on alliance contract so you just arrive wherever we are based, accept a contract and undock.
You can build the ships the alliance needs. If you are unexcited by PvP then you can run trading empires, indulge in regional arbitrage or make a fortune hauling or importing goods to where they are needed.
You can use the new Planetary Interaction features to play, well, Farmville in space, logging in each day to run industrial installations on the surfaces of planets. Or you can set up a complex series of installations to create industrial components others use to make stuff; or explore constantly-shifting wormholes to uncharted space; or build the expensive new T3 cruisers. CCP's goal is that virtually every aspect of the Eve economy be player-run, eventually, and this leaves gaps in the market for the savvy entrepreneur to exploit. Acting on the recent advice of economically-minded alliance members, for instance, many goons recently joined the ranks of the super-rich following the Tyrannis patch.
Applications will require a sponsorship key from a recruiter. You should have been given one of these already; if not ask your recruiter. Your application will be processed by our team of magic robots, and will usually process in 24 to 48 hours.
Most applications will require some form of security deposit, as calculated by the risk of your account being a spy. This deposit (if any) will be refunded in 30 days of non-spy behavior, or if you choose to leave us earlier (Please don't leave us, we love you!)
We offer many programs that make your life a lot easier.
This is only a small sample of the the kind of support you'll get in Goonwaffe. We feel that a corporation exists to support its players, and we're willing to back that stance up.