The most common mistake people make is camera shake. When you move the camera inadvertently at the time you press the shutter, you risk the chance of blurring your image or reducing the sharpness of the image. Keep it steady!
Most point-and-shoot cameras have a simple exposure override facility, normally allowing you to overexpose or underexpose your picture. So if the subject is predominantly dark, experiment by overexposing to compensate. If the subject is predominantly light, then underexposure is the way to go. Try taking a test picture, look at it on the screen on the back of your camera, check the histogram, and adjust your exposure compensation. Don’t be afraid to shoot four or five versions, as the LCD screen is not always accurate. You can delete the bad pictures later.
A very basic rule of composition is known as the rule of thirds, or the tic-tac-toe rule. Imagine your viewfinder or LCD monitor divided into nine equal-size squares, like a tic-tac-toe grid. Compose your picture with your subject center-positioned at one of the four intersecting points. This should help you compose more aesthetic portraits.