Single-camera production has been used as early as the 1960s. But the technique has become popularised recently by shows like "30 Rock," "It's Always Sunny in Philadelphia" and "Weeds." The main difference between single-camera production and multi-camera production is that the latter uses two or more cameras, filming from different angles simultaneously, while the former relies entirely on one camera to capture everything that's scripted.
The first and probably most disadvantageous aspect of shooting with single-camera production is that you only get one angle at a time. When you shoot with multiple cameras you're able to capture several things happening at once. This means that with single-camera production you have to attempt to get everything you want to happen in the right setting at the right time. If you cut a take and something interesting happens or someone nails their line, you're out of luck because there will be no other camera to pick it up.
Another disadvantage of using single-camera production is that you'll be spending a lot more time in the editing room. Because all of the film will be from one camera, all of your outtakes and mistakes will be have to be sorted out over time, as opposed to a multi-camera approach, where you can edit certain sequences, angles and scenes based on the camera that was shooting. More time in the editing room also means more time to add sound effects, transitions, music and other post-production processes.
Since single-camera production only uses one camera, of course, the director gets absolute control over every shot. If, for instance, you opted to go with the multi-camera production, you could get different angles and different takes at the same time, using different people manning different cameras. If one shot from the centre looks bad, you can always replace those frames with a shot from the right or left to easily and quickly patch things up. With single-camera production, you're stuck with the footage you get.