Roundabouts are one of those things that just seem to baffle Americans for whatever reason. They’re simple to use and far superior to stop signs at junctions: they’re much safer and improve the throughput of traffic.
Having nearly had many fender benders on the small, single-lane roundabout near my house, I felt compelled to give a simple explanation of how to use roundabouts.
In short, give way to traffic on the roundabout. Yield signs are posted at the entrance to the roundabout, which should be treated the same as anywhere else on the roads: yield to approaching traffic. Whenever you approach the roundabout, look left (true for the US, not for the UK and other places where one drives on the left) and if the roundabout is clear, you can proceed.
As you traverse the roundabout, other vehicles at entrances ahead of you should stop to allow you to proceed since you have the right of way.
Multi-lane roundabouts are a little different, but they’re pretty simple in the States because Departments of Transportation tend to add lane markings and direction arrows throughout, so they’re pretty foolproof. This simple animation from MDOT shows which lanes to use and when to yield.
In short, give way to traffic on the roundabout and don’t stop on the roundabout to let others in.
