It's not necessarily the same. A ring road is exactly what it says : it's a road in the shape of a ring, encircling an entire city. For example, the 'Périphérique' around Paris, or the 117-mile London Orbital (M25). A 'bypass' enables you to bypass ( i.e. not drive through) a certain area. For example, if a main road goes straight through the centre of a town, the residents may well want a 'bypass' to be built. This would be a stretch of road, maybe to the north or the south, which would join up with the main road on either side and allow traffic to cross the region avoiding the town itself. In other words - a ring road is a type of bypass ( in that it enables you to get from the north to the south of the city without going through the centre) but a bypass is not necessarily a ring road. A bypass is simply a stretch of road with a certain function. It's only a ring-road if it's circle-shaped.

SuKi has explained it well. I'll just add that we don't use "ring road" in the U.S.