Digital noise is a series of (more or less) random low/high pulses. Analog noise is an analog signal without any (apparent) order. There exist several types of noise, the most important ones being "white" noise, "pink" noise and "brown" noise.
http://www.triplespark.net/elec/oscillator/DWNoise0/
http://www.web-ee.com/Schematics/Pink%20Noise/pink.htm
The noise produced by this circuit has a flat spectral distribution and can be considered white noise. However, instead of having a gaussian output characteristics, it is uniformly distributed. A snap shot of the output appears below.
http://www.web-ee.com/Schematics/Noise%20Generator/noise.htm
http://www.web-ee.com/Schematics/TinyWhiteNoise/TinyWhite.htm
The circuit is an implementation of a flicker noise generator described in NBS technical note 604, Efficient Numerical and Analog Modeling of Flicker Noise Processes by J.A.
http://www.techlib.com/electronics/pinknoise.htm
Here is a circuit board for this. I made it skinny so I could fit it onto another board I was making that had some room left. When I was building this I noticed that I had to increase the gain of the circuit so I reduced the value of R3 to 500 ohms. I also buffered the outputs as you can see from the board.
http://www.musicfromouterspace.com/analogsynth/rayspinknoise.html