What does pixels mean when it comes to digital cameras?

Pixels are the smallest piece of anything in an image.
If you open up Paint if your on a PC, select the pencil tool, and zoom in as much as possible.
Click to make a dot, and zoom out.
That dot is one pixel.
The way an image works, is it arranges pixels of colors in certain patterns, that make up the image itself.

Essentially, a digital photograph is a mosaic comprised of pixels of color representing an image as accurately as possible, where a film photograph lets in the image exactly as the lens sees it.

To add to what Iscariot said, the number of pixels (obviously) determines the quality of the photograph or digital image in question. That's why most cameras today will come with some sort of indication as to what value of 'megapixel' that they photograph at. The highest I've seen personally is 21 megapixels but most retail digital cameras will be below 12 megapixels.