Mom is Unpredictable
For your mother to go between such extreme emotions of acting like your best friend and then threatening to kill herself and saying cruel things to you suggests that she is dealing with some kind of untreated mental health issue. Mood disorders, such as bipolar disorder, and substance abuse issues are just two of a few possibilities of what could be at play here. However, no matter what the specific issue is here it seems as if two things are clear: she is not getting the help she needs and you do not have the help and support that you need to deal with what is going on.
TeenHealthFX suggests that you start by discussing your concerns with your father. It would be helpful for him to encourage your mother to schedule a consultation with herself with a reputable mental health professional, such as a clinical social worker, clinical psychologist or psychiatrist. It would be most effective for your father to raise this with your mother at a time where she seems more peaceful and calm and would most likely be more receptive to what he has to say.
In addition to your mother being evaluated, it would also be helpful for you to meet with a reputable therapist as well for some combination of individual and family therapy. It is important that these family dynamics get addressed, and it is also important that you have someone you can talk to so that you get the guidance and support you need to deal with whatever is going on with your mother.
If your father is not receptive to encouraging your mother to get help and/or does not get help for you, then it would be important to talk to a trusted adult about this so that someone can intervene on your behalf (as well as on behalf of your mother). You could speak to your family doctor, a school nurse or school counselor, or anyone else who could intervene on your behalf. If a doctor or school staff member is unable to convince your parents to address these issues, that person can decide at that time if it would be best to involve your state child protective services.
If you live in northern New Jersey and need help finding a therapist you can call the Access Center from Atlantic Behavioral Health at 888-247-1400. Outside of this area you can log onto the US Department of Health and Human Services Substance Abuse and Mental Health Services Administration website for referrals in your area. You can also contact your insurance company to get a list of in-network mental health providers or check with your school social worker or psychologist to get a list of referrals in your area.
FX would also like to add that if you are ever concerned for your immediate safety if she is in a place of threatening to harm herself you can call 9-1-1.