In simple terms the camera sensor is the image capturing device of a digital camera.
The sensor captures light entering through the lens when the shutter is opened and enables it to be turned into a digital file to create your image.
The sensor is a silicon wafer that has been etched by a laser to create the millions of pixels. A single pixel is essentially a microscopic 'bin' that collects light on all its sides. The sensor itself can be considered an analog device. Microscopic electrical connections collect the light data from the bins and turn them into a digital signal which, when fed through the cameras processor, then forms the final image.
Larger sensors tend to produce better image quality in a number of ways due to the larger size and light gathering capability of each individual pixel. When combined with the capability of the cameras image processor this can result in better low light capability, dynamic range and overall image detail.