What is a circling approach?
To add to the above points...
Circling Approaches are used when the Approach you want to use and the runway you want to use are not aligned with each other. Say for instance, you are arriving to the terminal area from the North. The runway you want to use is from the South (probably due to wind). You want to get down to the ground ASAP. You can get cleared to approach the airport using an IAP with a Southbound Final Approach Course. They will also clear you to circle the field in a certain direction. We will use East in this example. You will then get clearance to land on the Northbound (from the south) runway. A play by play would be to:


Follow the full procedure for the IAP with the 180° Final Approach Course.
Once you reach your Circling Minimums, maintain you’re altitude, and turn 30° to 60° to your left. 
Turn back to your 180° heading in time to stay no more than 1.3 miles away from the runway surface for Category A aircraft (unless the airport has published alternate circling distances). It would be similar to a (right in this case) downwind leg
Parallel the runway to a point where you can make a normal downwind to base turn.
Start your descent to landing
Land on runway 36


If you never get the runway environment in sight, go missed at the original MAP.
If you lose sight of the runway environment at any time during your circling maneuver, immediately circle back over the runway to fly above it on your missed approach course. And go missed. Remember, the safest place in the airport environment is directly above the center of the runway.

One thing to be aware of on Circling Approaches is that the Circling Minimums are typically below Traffic Pattern Altitude. You will be flying slower and closer to the ground in the airport environment than you are used to. Caution must be exercised. 

Also, the Circling Approach does not have to be a complete 180° turn around to land. It could be in any direction.

Finally, some approaches only have Circling as an option. This is due to the approach itself not being aligned with the runway. In the example below, the runway is aligned 179° and 359° magnetic.