Who decides how long will I have my braces on? How is that determined?

Treating doctor! The treating doctor (an orthodontist i hope) will determine the time treatment will take. It depends on many things, especially the severity of the malocclusion and the quality of the clinician. Normal full orthodontics takes anywhere from 14-24 months usually. Good luck.
You and your doctor. Your orthodontist can give you his/her best estimate as to how long your treatment will take. Their estimate is based on the severity of the malocclusion. Once you have started treatment, the length of your treatment will be influenced by how well you brush and floss and also by how well you avoid forbidden foods, etc., that may break your braces which will slow down treatment.
Orthodontist. Babyi, when you go for an orthodontic consultation, the orthodontist will assess your orthodontic needs and determine treatment options and treatment timing. The average orthodontic treatment takes 12-24 month depending on the complexity of problems.
Rx & compliance. 2 factors affect your time in rx: 1. The complexity of your malocclusion. The more difficult to treat, the longer in treatment. 2. Your compliance. Treat the instructions from your orthodontist as rules, not just guidelines. The better you cooperate, the less time in rx, the better the result. And make sure you seek treatment from a university trained qualified orthodontic specialist.
The orthodontist. The duration of treatment is based on a number of factors including the difficulty of the correction, the amount of tooth movement, what type of orthodontic biomechanics are required, and what types of appliances and devices are needed. Ask your orthodontic specialist to assess your needs and provide an estimated treatment time.
The orthodontist. Based on what movement is required. Length also varies based on you-- your care of your teeth during treatment and if you where rubber bands (if requested by the provider).