Dust on the sensor can be a problem with digital SLRs, and it manifests itself as diffuse dark spots which appear in the same place on each image. Compact digital cameras don't suffer from this problem because they don't have interchangeable lenses. The lens and sensor are sealed units that don't allow dust to get in. Digital SLRs do have interchangeable lenses, and dust can get in and settle on the sensor. Some models have in-built 'cleaning' systems which attempt to shake dust free, and these can be effective though don't necessarily provide a complete cure.There's a dust spot right in the middle of this picture, and while this can be cloned out easily enough in an image-editor, it's a nuisance having to do it time and time again.
Dust doesn't always show up as badly as this. At wider lens apertures, the reduced depth of focus makes spots paler and more diffuse, and if the background is detailed they can often go unnoticed. Sooner or later, though, you may decide there's no alternative but to clean the sensor, and this is a somewhat tricky job that's well within the average photographer's capabilities, but not really for the faint-hearted.