Well I can't ge too technical on it, cause I am not altogether sure myself, but I do know when to use it. If you have a strong magnetic field in the area around your monitor, such as an adapter transformer plugged into a wall socket, it will affect the electron rays in the tube. You will notice part of your screen warping and changing color. This field magnetizes part of the tube so the picture will be goofy. By hitting the degauss, the screen demagnetizes and goes back to normal.
LTJLover was right degauss demagnetizes the screen.
The light that draws the picture gets out of whack if it gets a magnetic field around it for some reason. Try holding anything magnetic up to your screen to see it. When you degauss a magnet is waved back and forth inside your monitor and and slowly pulled away from the light so it removes the magnetic field that has built up. Degauss will not mess up you monitor even if you do it every five minutes. They would not put a button there for everyone to push if it did. Some monitors degauss atomatically when you turn them on.
A CRT (Cathode Ray Tube) works by emitting a stream of electrons from a Cathode, which is situated at the back of the neck of the tube. There are three of these Cathodes (guns)one for the red gun one for the blue gun and one for the green gun. The electrons which are emitted from the Cathodes are attracted to the screen by a highly charged anode, which has a typical potetial of 24kV (24000 volts). The front of the screen is coated with red green and blue phosphors which glow when hit by a high velocity electron beam.
In order to make a picture this beam of electrons must be scanned across the screen. To do this we use a magnetic field, this is provided by the scan coil which is situated on the neck of the tube. There are a pair of coils for horizontal scanning and a pair of coils for vertical scanning. The magnetic field is varied in the coils to produce the scanning action. The scan starts from the top left of the screen (when viewed from in front of the set)to the top right of the screen then moves down a line and scans from left to right again until it reaches the bottom of the screen, it then returns to the top of the screen and stars again.
The intensity of the picture is effectively created by the video signal modulating the voltage on the cathodes which varies the beam current hitting the phosphor and so varies the light emitted from the phosphor.
The three red green and blue (RGB) electron beams that hit the front of the screen must hit their respective phosphors (ie the red electron beam must hit the red phosphor)or we will get beam landing errors ie if part of the red electron beam hit the blue phosphor then if you were viewing a person wearing a red jumper say then the jumper would look slightly purple as the red electron beam is also lighting up the blue phosphor. To help avoid this there is a shadow mask placed just in front of the phosphors which only allows the respective electron beams hit their phosphors.
Now to answer your question!!!
If you introduce an external magnetic field (say a non shielded speaker) which can influence the electron beam this will cause the electron beam to deiviate from its intended path and light up the other phosphors creating strange coloured patterns on the screen. A large part of the tv set is made from metal, including the shadow mask, which can Gauss up (become magnetic) causing beam landing errors. So a De-Gaussing circuit incorperated in all direct view CRT sets to remove any magnetism. This circuit creates a high strenght alternating magnetic field which decays to nothing and neutralizes the magnetic field of the metal around the set. This circuit is activated every time the set is switched on and is the buzz/rattle you hear during power on.
Part of the circiut is called a positor (positive temperature coefficient resistor (a WHAT!!!!)) which is a type of semiconductor resitor which increases in resistance as it heats up, as it heats up it increases its resistance reducing the current flowing in the de-gaussing coil (the other part of the circuit) which in turn creates the decaying magnetic field.
(A wire or coil will produce a magnetic field proportional to the current flowing in it)
once the de-gaussing circuit has been activated the positor will need to cool for about half an hour berfore it will de-gauss properly again. (Try manually de-gaussing and wait about 30 seconds and try again, you will see little effect the second time)
You will only need to activate the De-Gaussing circuit your self if you introduce a magnetic field near the set or if you turn the set around ( the earth magnetic field can gauss the set enough to affect the beam landing). In both of these cases if it has caused a problem the screen will seem patchy if you had a green desk top say then there would be couloured patches in it.
If you do activate it manually you will very brifly see the type of effect a magnetic field can have on a CRT, also you will see the decaying effect of the De-Gaussing circiut.
So effectively most people will never need to activate the De-Gaussing circiut manually as they dont move their monitors much or bring magnets near their monitors.
If you do see coloured patches on the screen then use the manual circuit to remove them, if you need to do this alot ie once a day then there is something causing you monitor to gauss up. I have never used the manual deguass on my monitor the atomatic de-gaussing is sufficient when it is switched on ( and I tend to move my monitor around the room a bit and only switch it on at the start of the day and turn it off at the end of the day).
May be if the monitor was never turned off you would have to regularly de-gauss the set (say every couple of days) but in normal use the manual de-gaussing will rarely get used.
Well its getting late im tired and ive got to be up in a couple of hours so im going to have a shower and hit the sack.
Hope this helps with your question.