The main two types of killers are killers for revenge, and killers for pleasure.
Killers for revenge, are either quick, impulsive, emotional kills, or planned out slow, suffering deaths, a lot of the time they might kill for a personal moral or religious belief, like John Doe (Kevin Spacey) from Seven or The Jigsaw Killer from Saw. Sometimes a person will be forced to spend a lot of time with a person they strongly dislike or believe doesn't deserve to live etc. Or someone might have wronged them or done something they didn't like so much that they believed the only fair punishment would be death, this person might think they are doing 'karma' or serving 'justice' and giving them what they deserve.
Killers of pleasure, like Jeffrey Dahmer. Are killers who kill simply because they enjoy it, or just want to. These killers are more likely to do something with the body after the kill or have some sort of ritual. Sometimes they will just have an urge to kill or a hunger that needs to be satiated. These killers are often more likely to be sociopaths, psychopaths or narcissists. This is sometimes due to a traumatic event in their life that they reacted badly to, but there are other reasons for people to become like this.
Killers for revenge can be sociopaths, psychopaths or narcissists, but a lot of the time they are impulsive kills and come from normal people. They can also be planned out though.