VGA CRT displays use magnetic deflection to move the electron beam across the face of the screen. That is why bringing a magnet close to the face of the screen will move the electron stream. You can also permanently magnetize the shadow mask so that you get an area that has weird colors, so don't place magnets close to your CRT screens.
A little known fact: keep all sensitive electronic devices away from the CRT when you degauss the screen. The degauss coil can induce unwanted voltages in devices that are either sitting on top of or directly beside the display, possibly causing damage.
As for LCD displays, they use what's called nematic crystals
http://electronics.howstuffworks.com/lcd.htm/printable suspended in a liquid. When an electrical charge is placed through the liquid, the crystals align themselves and the cell either blocks light or lets it through depending on the technology used. The crystals are not affected by magnetism.
As for placing you Axim close to a large older CRT display that has a screen acceleration potential of probably 30,000 volts on the faceplate (place the back of your hand close to the screen and watch the hair stand up!), you stand the chance of drawing a high voltage static charge which 'might' cause damage to the device. Not all CRT displays exhibit this though but why take chances. Large LCD displays are low voltage devices so there is no problem there.
As for flash memory, they work on retaining a charge in a cell that represents a specific data state. Since static electricity can damage any electronic device, always take the proper precautions when removing and inserting the parts into your equipment (ie: don't walk across a carpet in wearing nylon socks and pick up your Axim from the top of your CRT! Zzapp!);)
Hope this explanation helps some.