How does a typical psychotherapy appointment proceed?

Therapies differ. It depends on the type of psychotherapy. "dynamic" therapists encourage you to say anything that enters your mind (and don't speak much themselves). The therapist listens for patterns and underlying feelings, helping you understand yourself better. "cbt" therapists give "homework" exercises to help you un-learn unhealthy thoughts. Both kinds usually meet with you 45-50 min, once weekly.
Psychotherapy. Initially, there is information gathering from one's personal history. Mixed into that should be an atmosphere of security, genuineness, caring, and warmth.
The 50 min hour. The first few visits are allocated to obtaining a history. Then problems are determined and possible solutions are suggested. Each visit is usually 50 minutes. Each follow up appointment revisits the progress or lack thereof in accomplishing the previous sessions goals.
Talking. Usually the patient describes the emotional issues that he or she is suffering to the psychotherapist. The psychotherapist will do much more listening than talking. Occasionally a psychotherapist will ask empathic and relevant questions about how the emotional issues came into being, andhow they impact your life.