When you say that you are "alone" what you are implying is that there is no other person with you. For example, right now I am sitting all alone at my computer desk and typing away — there is no one else in the room. I am neither happy nor sad about it; as far as I am concerned, the fact that there is no one around is neither good nor bad. The word "lonely" on the other hand, refers to the state of the mind of an individual.
It suggests that the person is sad, feels unwanted, and is longing for companionship. The degree of sadness that the individual feels may vary. When one sits in a crowded room, one is not "alone", but one can still feel "lonely" because loneliness has nothing to do with physical proximity with other people. When one is "alone", one needn't feel lonely. Some people prefer to be alone.
On my own refers to when somebody lives by themselves without any help.