When it comes to griefing and player conflict, EVE is one of the most laissez-faire MMO games in the industry. Because so much of the conflict is player-driven and improvised, there are much sparser rules with regard to players bothering other players. Still, there are some things that can be done to minimize trouble.
How It Is
CCP is an interesting company, and they have turned EVE Online into a longstanding success that continues to earn money and grow over the long term, even as some other long-running games begin to falter and get left behind.
As things stand, the rules of EVE basically forbid dragging real life into EVE: no real-life threats, racism, or real-life information. There is also no hacking EVE accounts, stealing others' passwords, or any funny business involving real-money transactions. All that is just good business, and protects the game experience as well as CCP's soft, eminently-litigated underbelly.
It's all in the game.
Why It Is
It very much is not for everybody. On the EVE forums, nearly every page has a thread about some guy rage-quitting over something another player did. On sites like Ten Ton Hammer, anything dealing with the unseemly side of EVE usually gets a few comments along the lines of "just another reason not to play" or "now I know never to try playing." To players from other MMOs it all just seems so, well, hostile.
Over time, a lot of the mythos surrounding EVE Online has become about how dastardly players can be to each other. People that never, ever would play a science fiction MMO still avidly read the news about the game. Players that quit EVE still find themselves reading forums to keep current with the latest null-sec politics. PvP (whether in space, via the market, or through social engineering) is unmistakably the fuel on which EVE Online's engine runs.
In order for our stories in EVE Online to be meaningful, there need to be lasting consequences from them. That means we play for keeps: ships blow up and are gone forever, GMs will not reverse transactions for you, and really experienced-mercenary corporations can declare war on your rookie corporation just for kicks. Like the man says, that's the price you need to pay for freedom. Again, it's not everybody's cup of tea.
The High-Sec Griefer Playbook (And Fighting Back)
Much of the bad stuff that happens to new players occurs because they don't know the finer points of a particular game mechanic. Reading guides helps, but every player needs to get a few unfortunate experiences under their belts in order to get into the swing of things.
Can-flipping is a common activity in more populated systems. Newer miners will often store their ore in "jetcans", ejecting ore and dragging more into the resulting container. This has a big advantage: you can pick all the ore up at once with an industrial ship, rather than warping to and from a station every time your cargo hold is full. Unfortunately, griefers will come and steal that ore right out of that can since it is open to anybody within 2500 meters. When they do, CONCORD will let you shoot at them. This is generally a mistake. Can-flippers generally have an angle they are working. That angle may be a surprisingly dangerous hauler (Badger II industrials are popular) or more likely backup waiting nearby or ready to log on in that same solar system. Once you return fire, anybody in the same corporation as that can-flipper can get involved, opening fire on you as they like. The better solution is to skip jet-can mining entirely and to anchor some giant secure containers in the belt where you mine, dragging your ore in there as you like.
This seems to be a bigger problem as time goes on and more people realize how easy it is. The idea is simple: a dangerous ship can blow up a covetor or other mining barge with one or two volleys, before CONCORD has a chance to show up. CONCORD will still blow the attacker up, but that is little consolation to the miner, whose ship wreckage is usually floating in space by then.
The best way to protect yourself from suicide gankers are as follows: mine or run missions in a less populated area, fit at least some tank on you ship no matter what it is, and if a dangerous ship shows up in your belt or mission then you should leave immediately. Running level four missions in Motsu in a faction raven is a lot less safe than running the same missions somewhere out of the way in a less conspicuous ship.
War Is Hell
Declaring war on another corporation can be a big problem or no problem, depending mostly on what kind of corporations are involved. For our purposes the most likely case is that of dangerous pirate corporations declaring war on newbie-friendly organizations that generally don't understand how to cope with war. The ways to cope with war are essentially:
Fight back: This is generally the least likely option, since your enemies have probably chosen your corporation precisely because of your corporation's inexperienced nature. While it could theoretically be a valuable experience for you and your friends, it will more likely get you mad while your enemies savor your pain.
Hire help: Hiring mercenaries sometimes yields good results in war, especially against smaller or even one-man corporations. The down side of this is cost. You and your friends are unlikely to have deep pockets unless you buy a lot of PLEX. Still, some players have reported very good results with these.
The best solution to getting griefed is to ignore it and hope it will go away. It's the opposite of medical problems, essentially.
Negotiate with your enemies: The people that declared war on you probably have a reason to do so. Usually it is to extort ISK, though just as often it is for some cheap kills. It is possible that the attackers want a relatively small amount of ISK or some other concession. If nothing else, hear them out.
Break your corporation up: The last resort is for everyone to leave your corporation, breaking it up. If there is no corporation, there can be no war. You can form a new corporation later (though not right away, or some GMs might have a word with you for evading war mechanics). In the meantime you can coordinate your friends in an in-game channel. This is sour grapes for a lot of people and pretty disheartening, but it can be better than having your members get blown up every day.
More than anything else, don't feed the trolls. If someone screws you in some way, don't give them the added satisfaction of knowing that you're mad. Don't make impotent threats about revenge or how powerful your allies are, either. The griefers have heard it all a hundred times, and probably enjoy it. Don't give them the satisfaction or even worse, keep them interested in you as a target. The best weapon you have against them is boredom. If you can make harassing you boring by blocking the guy and not responding, you have won at least a moral victory. It's like dealing with trolls in any other internet medium: ignore them for long enough, and they will go away.