The short answer is, bigger is better! The image sensors of the various digital camera types vary in size. The so called full frame image sensor is a copycat of the image size yield of the esteemed 35mm film camera. These image sensors measure 24mm height by 36mm length. The full frame size is often called FX in the jargon of the industry. Other camera types sport a smaller image sensor fashioned after a film format known as Advanced Photo System. These sensors measure 16mm height by 24mm length. The industry jargon for this smaller senor size is APS-C (the C is for classic format) often this format is called compact digital or DX.
The size of the image sensor somewhat relates to the most important factor which is pixel count. As a rule of thumb, the higher the pixel count, the higher the resolution. However, picture quality is also affected by the actual size of the photosite (pixel). Larger photosite likely deliver a higher resolution picture.
That being said, as to sharpness and resolution, much depends on how you view your pictures. The typical computer monitor and or TV display a lowered pixel count. Picture taken with camera with a super hight pixel count and then viewed on a typical computer monitor, suffer in this regard. This is true because most likely the software of the viewing system will discard a high percentage of the pixel count.
While pixel count and sensor size is important, it is likely not as important as you might think. Time marches on and todays compact cameras are quite satisfactory for most imaging tasks.
