'Megapixels' refers to the number of pixels (in millions) captured by the camera's sensor. It's still used as a principal selling point for digital cameras, even though it's no longer anywhere near as relevant as it used to be and, in some cases, more megapixels do more harm than good.
In the early days of digital photography, cameras didn't have many megapixels. This limited the size of the pictures you could print before the blocky pixel pattern became visible. It also meant that cameras couldn't resolve much fine detail.
So higher megapixels meant bigger, smoother prints and finer detail, but only up to a point, because the other limiting factor is the physical size of the sensor.
Compact digital cameras use very small sensors. There's a limit to how much lenses can resolve on tiny sensors, no matter how good they are. And the smaller the sensor, the more you have to blow up the image to get a same-sized print. We'll call this the 'enlargement factor'. With enlargement factors over 30x, you can expect to see the image quality deteriorating, no matter how many megapixels you've got.
The table below should make this clearer. It shows how many megapixels you need for different-sized prints, but it also shows the enlargement factor required for these print sizes with different types of sensor.
As you can see, the small sensors in compact cameras are hitting very high enlargement factors very early on. This is what limits their picture quality in big print sizes, regardless of how many megapixels they have.
In fact, higher megapixel ratings bring serious technical difficulties which do affect the picture quality. If the megapixel count goes up but the sensor stays the same size, it means the individual pixels ('photosites', to be exact) on the sensor have to get smaller. This makes them less sensitive and more prone to random noise. As a result, makers have to use strong noise-reduction processes to make the picture quality acceptable, and today's high-resolution compact cameras often display weak definition and a 'smoothed-over' look to subtle textured details. The pictures you get are no better than those from cameras of five years ago with half the megapixels.
So why do makers keep increasing the megapixels? It's because megapixels sell. Most buyers will be unaware of the technical implications and just see a bigger number as being 'better', and as long as that keeps happening, the makers will keep on increasing megapixels regardless of whether it's a good thing or not.
To sum up, increasing megapixels in compact cameras are achieving nothing except camera sales. In fact, in many ways it's driving picture quality downwards not upwards.
Hybrid cameras and SLRs are different because the sensors are much larger and they haven't quite reached the same level of 'megapixel saturation'. Even here, though, sensor size counts for more than megapixels, and you shouldn't necessarily expect a proportional increase in resolution just because one camera has 18 megapixels, for example, and another has 12.
