A dark patch, area or other discoloration on the ceiling is a sign of a water leak. These spots are often a sign of a roof leak, but if you have an upstairs bathroom above the area, the leak may be coming from that bathroom. You will likely not see signs of a leak on the ceiling below a bathroom until the leak has been going for a while, so you should turn your water off immediately to limit the damage and then call a professional plumber who offers water leak detection and has the ability to fix it.