I have a fracture in my right hand. Can I still play golf with my friends?

When healed. I wouldn't recommend playing while you are in a cast as this may cause problems with the bone healing properly. Not to mention your game will be affect and not for the better. If you must go out with your friends to the golf course just volunteer to drive the cart.
Depends on injury. Depending on which bone was broken, and how severely, recovery can be a few days to a few weeks. So much depends on which bone, where the bone was broken, and whether treatment is needed (surgery vs splint vs cast), that it is hard to give an answer in this short space. See your doctor, get x-rays, and get referred to a qualified hand surgeon, if your doc thinks you need it. Ask him first.