Digital Noise has the appearance of colored speckles that look somewhat similar to "grain" from high speed film, or when you make a poster sized enlargement.  Digital noise is common in low light situations and in dark skies taken sunset or sunrise.  Basically, the sensors in the camera get a little confused in low light situations.  Digital noise is present in the Low Light editing example, so I am simply showing some of the images from that page.

This candid photo was taken beyond the range of my flash, thus it is terribly under exposed.  One of the morals of this example in the Low Light Page is that severely under exposed images can be enhanced significantly.  Fig. 1 is the original, Fig 2 shows the enhancements, and Fig. 3 below shows the digital noise on her face.  Fig. 4 shows the results of removing the digital noise using some software called Neat Image.
 

Fig. 1.  Original

Fig. 2.  After changes in Low Light Page

Fig. 3. Close-up showing Digital Noise

Fig. 4 Noise "removed" from her face only using Neat Image

Photoshop CS and Photoshop Elements do not have effective tools for removing digital noise.  Most introductory books on these software packages will tell you how to remove noise with the two Photoshops, but the techniques are not effective because they simply blur the image.  The problem is that they blur edges that you want to be sharp.  If your image has no sharp edges, then their blurring techniques are OK.  Generally, you will get an image that looks slightly out of focus.  One place where blurring can be effective on noise is when you select only the blue areas of the sky where there are no edges and noise is very common.

Some good third party software for removing noise include: Neat Image, Grain Surgery, and nik Dfine.  These can be loaded as plug-ins with Photoshop and they cost between $50 to $150.  I have had good success with Neat Image.  Most casual photographers don't really need to deal with noise unless they plan to make enlargements in the 8" by 10" range, or larger.

To minimize noise in the first place, adjust your camera to the lowest ISO setting possible.  That might mean that you need to deal with manual settings.  I try to shoot most all shots at 50 ISO if possible.  This is similar to shooting "slower" film with less grain.