I had a operation to repair a tendon on my hand. When will I be able to know whether the operation was successful?

Therapy is key. In addition to the mechanism of injury and the quality of the surgical repair, physical therapy is very important in the managment of a tendon repair and will help to determine return to function.
3 months. If the surgery was performed to repair a lacerated tendon (from a kitchen knife, for example), it will frequently take 3 months from the date of surgery to gain most of your flexibility back. It will often take 6 months from the date of surgery to maximize your flexibility. You should be able to judge the "success" of surgery at about 3 months based on your progress.