I have a point and shoot camera that has a very nice zoom, and has the usual image stabilization etc. When I press the button to take a picture at, say, a concert (low light, moving people, pretty much the worst scenario) the picture on the LCD looks fine. A little grainy, but not bad. Then the camera thinks for a second, and makes the picture blurry. It almost seems like it takes another picture right after mine, one that pulls in more light, and then combines the two. So I get more light, which is nice, but the picture is very blurry because the subject was moving. Any idea why this is happening?
If you are shooting by looking at the rear LCD or an Electronic Viewfinder (EVF) to frame your shot here is the most likely scenario without more information from you added to the question (Camera model, specific settings, etc.).
The LCD screen (either on the back or inside the EVF) is using the sensor to produce a series of pictures much like a video camera, one right after the other for you to view the scene and compose your photo. It is probably doing it at somewhere around 30 frames per second. It can 'get away' with this even in low light for several reasons. The first reason is that the small LCD is very likely much lower resolution than the native resolution of your camera's sensor. When a high resolution image is reduced to much lower resolution, the amount of image noise is also reduced as adjacent pixels are combined or averaged. The second reason is that the random nature of noise also means from one frame (that is streaming to your LCD at 30fps) to the next the noise moves around and your eyes perceive less noise than is probably present in each individual frame. Remember when VHS VCRs first came out with freeze frame and slow motion playback? You couldn't believe how blurry each individual frame was when viewed statically or at a very slow frame rate, yet when played back at full speed your eyes and brain are able to take that information and reconstruct a much higher quality picture than each individual frame contains.
But you want the picture you save to be at the sensor's actual resolution, and you want it to have as little noise as is possible. In order to do this in low light, your camera has to record the light hitting the sensor for longer than the less than 1/30 sec it uses to produce the video streaming to the LCD. What it typically does is freezes the image displayed on the LCD while it collects the light for your picture on the sensor. It then reads the data off the sensor, sends it to the camera's processor, the processor converts that data to a preview image thumbnail (among other things), and shows the thumbnail on your LCD screen.
So that first still image you see, that you describe as "a little grainy, but not that bad" is basically a low resolution video frame grab of the last frame streamed to your LCD before the camera starts to take the picture. The 'frame grab' is displayed on your LCD while the camera actually takes the picture and processes it. Then you see a reduced preview version of the picture you actually took and saved to the memory card. The reason it looks so blurry in low light is that the required shutter speed is so long that anything moving in the scene is recorded as a blur.
My guess is that the lower resolution on the LCD is hiding the fact that there is a lot of noise due to (very) high ISO for the live view pictures. The first parameter you should look at to figure out why a picture is blurry is the shutter speed/time.
In concerts and other indoor dimly lift venues there is usually very little light. Your camera have to compensate for the lack of light with high ISO, large aperture (small number) and longer exposure. It's the longer exposure that most likely is the cause of the blur you seen in your pictures.