Security status is an attribute of every star system in New Eden. Security status goes from 0.0 to 1.0 and basically represents the degree to which CONCORD (the police force of the eve universe) keeps crime in check. It affects both PVP and PVE play.
The lower the security status of the system the tougher the NPC baddies who will show up and the more ubiquitous they will be. In 1.0 and 0.9 space, NPC baddies will basically only show up in missions or other designated combat zones (dungeons, if you will... more on that another time); they will never randomly jump you. In 0.5 to 0.8 space you start seeing them show up randomly at asteroid belts to harass miners (or to be blown up by players who have gone to the belts for the express purpose of doing just that). Of course the baddies will be tougher in the 0.5 systems than in the 0.8 systems. once you go into 0.4 or below you'll start seeing baddies showing up everywhere: in asteroid belts, at stargates, sometimes even laying ambushes right outside of stations. Also, the lower the security status of a system, the more valuable the asteroids in it will tend to be and the better the reward payouts will be from the agents who make their homes there.
The rules of PVP engagement follow a couple of solid cutoffs relating to system security status. In 0.5 or above, if you shoot another player character unprovoked, the CONCORD fleet will show up shortly thereafter and completely obliterate you. CONCORD has more than enough firepower to destroy the most powerful player ships in the game. There is a 100% mortality rate for CONCORD fleet attacks. 0.5 and above is commonly referred to as High Sec.
In 0.1 to 0.4 (Low Sec), if you attack another player unprovoked you will get a red "global criminal countdown" in the top left corner of your screen for fifteen minutes. This means that CONCORD is out to get you until that timer runs out (they have short attention spans). Fortunately (or unfortunately, depending on your viewpoint) CONCORD's fleet doesn't venture into low sec, so you won't be immediately set upon. They do however control sentry guns at all the stargates and stations in Low Sec, and if you venture within a 100km or so of any of the sentry guns while your global criminal countdown is active they will unleash their fury upon them. The sentry guns do more than enough damage to instakill most smallish ships, however there do exist plenty of player ships that are capable of tanking sentry gun damage, so don't feel too safe in low sec just because there are sentry guns nearby.
In 0.0 (Null Sec) CONCORD has no presence at all. It's every man for himself and the only consequences for your actions are the ones that other players enforce upon you with their guns. Null Sec is the only part of space where player organizations can officially claim sovereignty and so it tends towards large scale fleet warfare, whereas PVP in Low Sec tends more towards solo piracy and small gang skirmishes.
New Eden (the eve galaxy) is basically setup with a large nugget of High Sec in the middle surrounded by a small buffer of Low Sec and then a thick outer ring of Null Sec. And then there's wormhole space which is huge and uncharted and entirely Null Sec (but with no player
It's tempting to think of High Sec as being a No PVP zone and Low Sec and Null Sec as the PVP areas, but that's really not the case. As I've seen it astutely put in the forums: "There is a 'flag for PVP' button in eve. It's labelled 'undock'".
There are a number of ways that PVP can happen in High Sec:
Suicide Ganking: CONCORD punishes crime, they don't prevent it. The lower the Security Status of the system, the longer it takes CONCORD to show up and reprimand the offender. In 1.0 space, they arrive almost instantly, but in 0.5 it can take up to 90 seconds, which is more than enough time for a properly fit attack ship to destroy a cargo hauler or other flimsy vessel. Obviously the cost of this maneuver is high, so it doesn't happen very often, but it does happen (The general scheme involves one person in a hauler equipped with a cargo scanner and a second player in a cheap battleship. Once the hauler identifies a ship with a cargo load that is worth significantly more than the battleship is, the second player ganks the victim and then--after CONCORD has wrought its vengeance--the first player loots both wrecks.)
Can Flipping: Theft is not considered a real crime in the eyes of CONCORD, but they tacitly approve of vigilante theft prevention. If you kill an NPC or jettison a canister, then the contents of that wreck or can are considered yours but anyone can come along and steal from them. If they do, then they will get a yellow "aggression countdown" towards you for fifteen minutes and, for that time span, they will appear red in your overview. This means that you (and other members of your corporation) can shoot at them freely without CONCORD intervening. However, once you fire the first shot, they are allowed to shoot back.
War Declaration: A player corporation can declare war on other player corporations, thus giving those corporations free reign to fire upon each other anywhere regardless of Security Status. The corporation declaring the war has to pay a hefty weekly "War Maintenance Fee" (read "Bribe") to CONCORD to keep them looking the other way, so unless you really piss someone off or give them reason to believe that you are a lucrative target this is a relatively minor danger. Also, when you first join the game you start in a rookie corp which can't have war declared on it, so you don't have to worry about this mechanic right away anyhow.
The final thing that system security status does is dictate which parts of space pirate players may not enter. Your character has a statistic called a "security rating" (Unfortunately, this is occasionally referred to as "security status" within the game, but it's easy to keep them straight. Star systems have a security status, people have a security rating.) You start with a security rating of 0 and it can vary from -10 to +10. Each time you gank someone in high sec or fire upon them unprovoked in Low Sec your character's security rating will decrease significantly. Each time you shoot an NPC pirate your security rating will increase slightly. If your security rating ever drops below -2, you will be banned from 1.0 and 0.9 space until you kill enough NPC baddies to bring your rating back up. As your rating drops further you will be progressively banned from more and more of high sec until, at -5, you can not enter high sec at all and are marked permanently red to everyone even in low sec (meaning that if they blow you up they actually GAIN security rating rather than drawing CONCORD's ire).