Depression: Tell the person suffering from depression that you've noticed that they seem down or depressed lately. (Unless they have already told you that they are suffering from depression.) If so, tell them that since you now know that they're sad that you want to help. This will be very re-assuring to them. Sometimes the depressed don't even feel like telling anyone because of the stigma associated with depression. Also, make them feel like they are needed. Talk to them when you are upset about something or when you want to vent. Trust them with things that you wouldn't tell just anyone else. This makes them feel very important to you. Depression lies to them and makes them feel like they aren't needed by anyone. This acknowledgment that you know how bad they feel may be the little "push" they need to start talking about it, and maybe even seek help.
Eating disorders: First educate yourself about the eating disorders. Let them know that you are trying to understand and that you are there for them. Try to understand that although you know the eating disorder is hurting them, they percieve it as helping them. They will need a lot of patience and support from you to recover. Try to get them to talk about why they started, this might help them stop.
Anxiety: If your friend is panicking, just try to help them calm down. Some ways that help are deep breathing, meditation, drawing, etc. Dont become impatient if the distractions seem to not help, anxiety can take a while to calm down, especially if the person doesnt realize they are panicking.
Abuse/rape: Listen to what your friend is saying. Don't judge whether it was really abuse/rape or not. Dont tell your friend he/she is over reacting. Suggest that he/she reports who hurt them, but don't make them report it. Just let them know that you are always here for them, they are never alone, and no matter what they think it was not their fault. Keep telling your friend that they didnt deserve what happened and they didnt do anything wrong.
Suicidal thoughts/attempts: Let your friend know that you are there for them no matter what. Let them know that they are worthwhile, that life gets better, and that they are not alone. It will be hard, and it wont happen over night, but eventually you being there will help them.
Self harm: Let your friend know that you are there for them. Don't get mad if they relapse, but try to help them recover. They cant recover on their own so try to get them to see that. They might fight you even though you are trying to help them,but dont take what they do/say personally, its the addiction talking.
PTSD: If you notice your friend slipping into a flashback or having a nightmare, dont touch them, just try to talk them out of it. Tell them that its not real, that they are safe now, that what happened isnt happening anymore. Help them calm down by distracting their thoughts. try to get them to focus on what is really around them by having them say out loud 5 things they see in the room, 4 things they feel, 3 things they hear, 2 things they smell, and one thing they taste and repeat that until their mind is able to focus again.