Types of Yandere
In the west, yandere is basically always associated with the jealous, violent type of yandere, which is the most common. However, given that the word just means "mentally sick and in love," there are other types of yandere which would be recognized only in the Japanese speaking community.
A single yandere can (and normally will) fit in multiple of these types at a time. In particular, Yuno Gasai fits a dozen of them, which is why she's an ideal example of yandere.
mugai-gata (無害型)
Harmless type.
"As long as he/she is happy, I'm happy."
The most harmless type of yandere, the one where they have fallen in love in an unhealthy, obsessive way, but they don't do anything insane about it. They will try hard to become your lover, but won't harm anybody in the way. If you get into a relationship with someone, this type of yandere won't attack you two, they'll be happy you found happiness, but perhaps they may still hope that you'll choose them instead one day.
This can be associated with Megadere, Undere or Dorodere. Harmless type of yanderes, who could also be one or three of these three types and would not even come close to being a real yandere.
They can either be viewed as a crazy, obsessive version of a Deredere or a sweet, loyal version of a yandere.
kanchigai-gata (勘違い型)
Wrong idea type.
"He/She gave me an eraser... this must mean he/she loves me!"
They get the wrong idea when you do something in order to match their expectations. If you say you don't love them, they'll think you're lying because you don't want to hurt them or something. If you do something out of kindness to them, they'll think it's out of love. And so on.
shuuchaku-gata (執着型)*
Obsession type.
"I sent you 50 messages, why didn't you answer? Where were you? What were you doing?"
Tries to learn everything about you. Personal information, hobbies, routine, etc. Sends regular messages to check on you, asks why you haven't answered if you didn't, wants to know what you're doing always. All the time. And, if possible, walks around with you all the time too.
The obsession type doesn't necessarily want to monopolize you. They'll let you hang around with friends, etc. but wants to know everything you do. They probably want to go with you, too. If they can't go somewhere with you, they might stalk you.
If they learn a person is too close to you, they might attack.
sutookaa-gata (ストーカー型)
Stalker type.
Follows you around, often without you knowing. May be in broad daylight or at night, when you're walking alone on the streets. Maybe even online. A person of this type is also often of the obsessive type.
dokusen-gata (独占型)
Monopoly type. (This is the most common type.)
"Who is that person you were talking to?"
They want to monopolize you. Will ask who you talk to and hang with, in extreme cases won't let you be with anyone else, not even your friends.
This type of behavior is unfortunately common in real life. It displays insecurities and lack of trust in the relationship which might develop into much worse yandere behavior.
ison-gata (依存型)
Dependence type. ("Dependence" as in addiction, alcohol dependence, chemical, etc.)
"No, don't leave me! Please! I can't live without you!"
Can't live without you. Begs you to not leave them or throw them away. Say they will die if you go away. May lose will to do anything if you aren't watching. If you do leave, they might go full crazy and end up doing something crazy like going on a murderous rampage.
touei-gata (投影型)
Projection type.
"You're just like him/her."
After their former love turned out to be a completely different person than they loved, got a girlfriend/boyfriend, died or something like that, and the yandere can no longer stay with them, they will search for someone who was just like what their love was, and it turns out that you end up being that person.
This type of yandere is very innocent at first, but if they're given power to dominate their new beloved, they might end up trying to make them more and more like what their old interest was. For example, wearing the same things, doing the same things, etc.
Sometimes, this yandere might project their ideal lover not on someone else but on the same person. That is, they might say "you are not him/her, he/she does this," to you even though you actually are him/her and you don't do "this".
shoushitsu-gata (消失型)
Disappearance type. (This has nothing to do with Haruhi Suzumiya)
"He/She will never love someone like me... why exist?"
They love you, maybe you know that, maybe you don't, but unknown this gives them a crushing depression. They think you'll never love them and they have no chance, but they can't stop their unhealthy feelings of love. They thinks you're too good for them, or that they're too worthless for you. This makes them slowly fade away, disappear from your life. Until they completely disappear forever (probably suicide).
This is an atypical yandere, since it has so little effect on other characters' lives.