Here, scroll down, just short of the bottom.
That shows what happens when a monitor is turned various directions. The color gets really forked up. Degaussing cures it, or at least it should. Now to finish what Michael started - a detailed explanation.
CRTs use a beam of electrons to excit phosphors on the front of the monitor. They are guided by electromagnetic fields. The electrons can also be affected by regular magnetic fields, such as a small magnet, or Earth's magnetic field. This makes the electrons hit the wrong phosphors, causing funky colors that'd keep a hippie occupied for hours on end.
Now, around the outside front of the picture tube are some coils of wire. Degaussing sends a burst of electricity through them. It can seriously be powerful - higher quality monitors have a more powerful, and longer degauss cycle than cheap monitors. My Nokia 445's degauss makes my 1250VA UPS beep for mercy due to the momentary current overload.

This surge of power "formats" the front of the screen, removing any local magnetic irregularities that may form in the shadow mask or aperature grille, whichever your monitor may have. This gives uniform color.
Now, if you do something stupid like take a poweful magnet and stick it to the screen, you'll probably need a professional grade degausser to REALLY blast the offending magnetic field out of the screen.