A digital photographic image is made up of an array of dots, called Pixels - reputedly from "Picture Element". Each pixel can be only one colour; the number of different colours available (the colour depth) depending on the bit width of the storage format or image sensor (such as a CCD).
The maximum possible sharpness of an image is largely determined by the number of pixels used to store, display and/or print it; more pixels means smaller pixels, which can capture increased detail. Similarly, a wider bit width allows more colour differences to be rendered, and so better subtlety of tone. The number of pixels (or, more properly, the number and bit-width/colour-depth) is often called the resolution of the image. More pixels (and greater bit widths) can mean greater possible quality, but also raised memory or disk storage requirements, longer processing times and slower transmission over networks. Too few pixels (or too shallow colour depth) can result in the pixels becoming noticeable in an image, which can show "jaggies" - jagged edges to lines and loss of fine detail.
A Megapixel is a million pixels - occasionally abbreviated to Mpx or Mp. Typical digital cameras have resolutions specified in megapixels.
The "native" resolution of a camera is determined by the resolution - or number of pixels - of the image sensor, although some cameras use software to interpolate their sensor images into higher resolution pictures.
Pixels in an image are usually square (i.e. the same width as height), and arranged in a rectangular grid. However, there are exceptions; some computer displays (e.g. the EGA "standard") have had rectangular pixels (e.g. taller than they are wide), and some image sensors have their pixels in non-rectangular grids (e.g. the octagonal pixels of the Fuji Super CCD). These exceptions are reasonably rare - and for most purposes are usually processed into square pixels once they leave the camera.