If you're in a player-run corporation, there's always the risk that another corp will declare war on yours. The enemy pay a small war fee and CONCORD agree to look the other way any time your two corporations fight. Members of your corp become valid PvP targets for the enemy corp in high security space and vice versa. If you ever find yourself on the receiving end of a war declaration, an eve-mail from CONCORD will appear in your corp's inbox explaining everything. The mail arrives 24 hours in advance of the war beginning, giving the victims fair warning and some time to prepare.
A useful war-time trick is to Google for the enemy corp's name and see if they have a killboard. Get the names of their pilots from the board and have all your members add them to their address books. This will make it easy to see if there are some of them online, which is a sure sign that they'll be running a roaming gang looking for war targets. The enemy killboard will also provide intel on ships and fittings they typically use, which can be useful if you intend to fight back.
The best thing you can do during an unsolicited war is to deny the enemy any kills. Many corps that issue wardecs are made for the sole purpose of engaging easy targets. Since they only have three war slots, they're likely to drop the war with your corp and find another to try. If all else fails, the option also exists to simply leave the corp, at which point you are no longer a valid target for the wartargets.
With the currently saturated mineral market, many Tech 1 ships cost barely anything to lose. After accounting for insurance costs and payouts, a Tech 1 fitted destroyer can cost as little as 100k to lose, cruisers under a million ISK and battleships only a few million. The unfortunate consequence is that this makes them cost-effective to use in suicide ganks.
Players will typically cargo-scan and ship-scan passing ships at a stargate and look for someone with valuable loot on board. If the ship is weak enough to be killed in one or two volleys from a group of suicide attackers, they'll open fire and have another character pick up the loot. CONCORD will warp in and destroy the suicide character's ships but by that stage it's too late. Having been on both sides of this process, I've picked up a few tips on how to stay safe from suicide attacks when hauling. Unfortunately, there's not much that can be done about people suiciding mining ships for fun.
Although some people suicide gank for sport, the primary motivation is usually to make a profit and so they'll want to hit valuable targets. To make yourself unappealing, carry no more than 20-30 million ISK's worth of goods in a Tech 1 industrial, 50-100 million's worth in a Tech 2 transport ship and no more than 1-2 billion ISK's worth in a freighter. It's also much safer to warp manually between stargates than it is to use the autopilot as most gank squads scan pilots as they approach the gate on autopilot. When warping manually, you'll land right within jump distance of the gate and so bypass most suicide squads.
Can flippers and loot thieves:
Ordinarily, you can't be shot in high security space without CONCORD coming to tear your attacker to shreds. If you steal an item from a container or wreck someone else owns, however, all bets are off. You become flagged to the owner of the container you just stole from and for 15 minutes they can engage you without CONCORD interference. Some pilots use this mechanic to trick people into becoming vulnerable to PvP, then blow them up. We've all seen the little cargo containers sitting outside Jita 4-4 with names like "free stuff" but there are some less obvious ways a player can be caught out.
The most common time when people fall victim to can flipping and theft is when mining. If you're mining into a jettisoned container, an aggressor may steal your ore. They may also create their own jettison container right next to it and drag your ore over into it. The theft flags them to you for PvP, so they flash red and you can attack them, but at this point they still can't attack you.
If the thief is in a small ship like an industrial or frigate, you might be tempted to attack but this will give them permission to retaliate for the next 15 minutes. Even if you do manage to blow them up, there's no telling whether they'll come back in a new ship with bigger teeth before the 15 minute timer expires. Taking your ore back from their container has the same effect, giving them permission to shoot you. Your only reliable option is to stop mining to a jettisoned container and call the stolen ore a write-off.
While high security space is undoubtedly the safest place to live in EVE, it's not without its perils. War corps, suicide gankers, loot thieves and can flippers stand ready to make life difficult and it's up to you to stay safe. Following a few simple safety tips and having a good feel for EVE's aggression mechanics will go a long way to keeping your hull in one piece.