Thieves and murderers can be killed by anyone.
Crimson Order and Knight Templars can only kill people who are wanted for crimes. You are not supposed to use your guild advantages to harm others, so be careful. If you accidentally kill someone, you could lose your guild standing! Always use the ID rod before defending yourself against other players. I know it's tough, but these guilds are supposed to be enforcers, not normal players who are out for themselves.
I'm sure the next question will be something along the lines of "why can't I defend myself if attacked?" Unfortunately, the melee scripts are hard coded. I would love to add a variable that keeps track of who attacks first, but I have never found a way to link a function call into the melee scripts. They are what they are, so be careful. If you do accidentally kill someone, you're allowed to repent and clear it one time. Plus, there's an item in the game that will clear your record if you deserve to get it.
What about innocent players who aren't murderers or thieves? If a player isn't wanted, he can only be killed by someone of approximately the same level or higher. Killing someone lower than you by more than a couple levels is considered an unfair kill, and you'll be labeled a murderer. Killing someone who is substantially lower than you (by 15 or more levels) gets your character sent immediately to prison, and the victim is respawned at no cost.