High security space:
In high security space (1.0 to 0.5 security rating), CONCORD police ships stand ready to punish anyone engaging in unsanctioned PvP. They take a few seconds to arrive but if you shoot someone you're not supposed to you can be sure they'll show up and destroy your ship. Engaging in unsanctioned PvP also globally flags your character as a criminal for 15 minutes, allowing anyone to attack you unhindered and signalling to CONCORD to destroy any new ship you undock with. What CONCORD don't get involved with is sanctioned PvP such as empire wars or combat with pilots that have been flagged as a valid target for some reason.
Players can become flagged as a valid target to you in many ways. They might steal something belonging to you or use remote repairers to aid someone else that's a valid target to you such as a war-target. Whatever the reason, those pilots that are flagged to you will appear with a red background on your main screen and overview. If you've committed any crimes that allow someone to attack you, the longest timer will appear on the top-left of the screen in yellow text. Moving the mouse over it produces a full list of all PvP flag timers for all your crimes. This also includes any pilots that have attacked you and any NPC factions you've been in combat with.
The primary way that someone will become a valid target to you in high security space is theft. If you steal something from another player's jettison container or yellow wrecks marked as theirs, you will become a valid PvP target to their entire corporation (or just them if they're in an NPC corporation). For the next 15 minutes, any member of their corporation can attack you without CONCORD intervention. Although this allows you to attack and potentially kill thieves, the flagging system is a double-edged sword. If you do choose to attack the thief, you then become flagged to him as a valid target for 15 minutes.
Don't be so hasty to attack a thief, however, as he may be fit specifically to take you down and even if you destroy him he can return in another ship. So before you kill that hauler that's stealing your ore or that frigate that's stealing your mission loot, remember that they have 15 minutes in which they can come back in a combat ship and kill you. For many that call themselves "high sec pirates", this is the ultimate goal of the theft. My advice here is to leave thieves alone or bring a corpmate for backup in a small ship like an assault frigate or a cruiser that can kill the thief. The thief will then be able to attack your backup but not your main ship.