There are some instances where you can legitimately shoot another playe in Eve, or be shot. The rules vary depending on where you are:

Hi-Sec:
You can shoot another player who has stolen from you, or has already shot at you, without alerting the authorities. In this case the player will appear as flashing red in your overview. The exception is if the player is in a pod. You cannot destroy a pod in hisec legally unless you are at war with that corp.

Do not confuse flashing red with having a red minus by their name (you, your corp, or your alliance does not like them), having a skull by their name (they have a bounty on them), or having their name coloured orange or yellow (they have a negative security status). If you are at war with a corp you will probably show the red minus by a players name but that does not mean that everyone with a red minus is a legitamate target.

Failure to adhere to these rules will result in a security standing drop and Concord showing up and blowing you up, pod and all, at varying speeds related to the security of the system (fastest in 1.0). You will not win against Concord. You can, of course, shoot someone expecting to be blown up for it, but you had better do a hell of a lot of damage with your first shot or it'll be fairly pointless.

Low security space:
Shoot anyone you like but you will take a security standing hit for it and if you are near some sentry guns you will get shot at for it (but you have better chance of lasting against guns than Concord).

0.0:
Shoot anyone you like, no sec standing hit, the only consequences are political.