First of all, I think you should talk to your mother about the conversation you overheard. Its possible that you misunderstood what she was saying. Before you jump to the conclusion that your own mother thinks that of you, you should try and talk with her about it first. Communication is key in maintaining any relationship even parent-child ones.
As for your problems with school and friends, depression is normal every time you are removed from what is comfortable and familiar to you. Give your friends a bit more time and they should start getting back to you. Their lives just might be as busy as yours is right now. When things calm down for them they'll start to call you and your friendship should return to normal.
However, if you feel your depression is more serious than what is normal for someone in your situation, talk to your mother about possibly sending you to a therapist. Have them evaluate you to see if therapy is all you need or if you should be referred to a psychiatrist and medicated for your depression.
Remember: No one need suffer alone. There is always help, all you have to do is reach out and take it.