Tailights burning

I have been replacing the brake light in the rear right of my car. I later got a new battery because the old one wore out. After I put the new battery on my car, now my tailights will burn or come on on their own. I wont have keys in or car on, won't even be around and they will come on. What do you think is wrong and approximately how much will it cost to fix. Thank you so much for your time.