One of the most important aspects of a camera is the size of its sensor. A larger sensor will capture more light, which results in improved low-light performance, dynamic range, and general image quality.
The more focus points the more flexibility in picking which part of the scene to focus on. They also give the image sensor a better probability in identifying the right area of the scene to focus on in more automatic modes.
The number of megapixels determines the resolution of the images captured with the main camera. A higher megapixel count means that the camera is capable of capturing more details. However, the megapixel count is not the only important element determining the quality of an image.
With a higher light sensitivity (ISO level), the sensor absorbs more light. This can be used to capture moving objects using a fast shutter speed, or to take images in low light without using a flash.
With sensor shift, the image sensor (rather than the lens) moves to counterbalance any vibration of the camera. This means the image will be stabilised regardless of what lens is being used.
Fast continuous shooting is useful for catching action shots.
With AF tracking, once you choose the subject and press the shutter release part way down, as the subject moves, the autofocus will follow it. No more out of focus shots.
Phase-detection autofocus is much faster than a contrast detection autofocus, allowing for sharper images.