Digital cameras have three basic modes: Off, Capture, and Preview. Capture is the mode you use when the camera is ready to record an image. Preview is when you're looking at your previously shot images on the LCD screen.
In Capture mode, the camera is ready to capture an image. On most digital cameras (except for many SLR types), there is a preview of what you're about to capture displayed on the LCD screen. It is important to understand that the process of capturing an image involves a few sophisticated processes most of which will be done automatically by the camera, but all of which you must initiate by first pushing the exposure button HALF WAY down.
When you push the exposure button half way down, you ask the camera to analyze you scene and make adjustments, such as auto exposure, auto focus, and auto white balance.
I have found that the best approach is to push the button halfway down first (to prepare the camera to shoot), then push it the rest of the way at the instant that you want to create a photograph. If you just simply push it down all the way in one step, you will likely experience a slight or pronounced delay or hesitation that is the result of the camera performing the above operations. This delay can be frustrating if you are trying to capture quick action.
! If the camera will not fire, it may because one of these above operations could not be performed, or had problems:
Preview mode allows you to look at the images you just shot, and can be a great tool to judge your photographic decisions before it's too late to make adjustments. Generally when you're in preview mode, you can use a set of buttons to shuffle through your images (forward, backward). You can also generally use the zoom control to allow you to inspect the images up close, or let you zoom out and see a set of thumbnails from which to choose. You can also delete images that you don't want to keep.