People often develop a close relationship with their therapist. You've been sitting in a room talking about very personal subjects, often with some frequency over time. Does this make you friends? Some people certainly expect that it does, but the therapist usually does not see it that way.
Psychotherapy is an asymmetrical relationship. The client opens up and the therapist generally doesn't. This is necessary in order to focus on the client's problems exclusively. How can trust develop in such a one-sided relationship? Since the therapist doesn't reveal nearly as much, the client will hopefully come to view the therapist as a safe, caring listener who is devoted to helping the client figure out their problems -- not the therapist's.
Friendship, on the other hand, is inherently two-sided. In most relationships, we open up gradually as the other person also opens up. As your friend, I know many things about you, and you know many things about me. We usually have shared experiences beyond sitting in a room, talking.
Therapy can certainly be a "friendly" relationship, depending on the personalities involved and on the therapist's theoretical orientation. Historically, certain psychoanalytically-oriented psychotherapists took pains to not reveal any aspect of themselves to their patients, believing that this would unhelpfully influence the patient's reactions (or transference) to the therapist. Most contemporary psychoanalysts and therapists, however, recognize that they are always revealing aspects of themselves, and that a goal of the therapist is not to hide their personality, but to foster the kind of relationship that allows for the fullest discussion and exploration of all the reactions that take place between the therapist and patient.
Your therapist probably won't be your friend because that would create a "dual relationship." Dual relationships occur when people are in two very different types of relationships at the same time. Many dual relationships are unethical in therapy. For example, it is unethical for a psychologist to treat a close friend or relative. It is also unethical for a psychologist to have a sexual relationship with a client.
One of the difficulties with dual relationships is that a problem in one relationship (such as a friendship or sexual relationship) can then cause problems in the other relationship (the therapy relationship). If you are mad at me because I didn't attend your party, it will be hard for you to open up in therapy. In addition to being a dual relationship, sexual relationships with clients exploit the power inherent in the one-sided nature of the therapy relationship. Such relationships are unethical on several grounds.
Can a friendship develop after therapy? While not common, it can happen. However, ethical guidelines frown on this for various reasons, including the idea that the transference aspects of the relationship, and the asymmetrical power differential established in therapy, never fully disappear.
If you are currently in therapy, expect your therapist to be someone who is easy to talk to. If he or she is friendly, this may be an added bonus. But remember that therapy is not the same as a friendship. By taking advantage of the personal and professional relationship that develops in therapy, you will be better able to make the changes that you want to make in your life.