It's interesting that no-one has mentioned why magnets do all sorts of horrible things to 'traditional' (i.e. CRT) television and monitor screens.
Firstly, let's consider how an old-fashioned black and white TV works. There is a large cathode-ray tube featuring an electron gun at one end, and a phosphor-coated screen at the other end. The electron gun emits electrons, which are attracted towards the electrically charged screen1; when they hit it, the phosphor glows.
Now, TV works by showing 25 or 30 still frames per second2, to give the impression of movement. Each frame is made up of many horizontal lines. To draw these lines, the electron beam has to be moved in a certain pattern, known as a raster scan. Consider when you are reading; your eyes move slowly along the line of text; then when you reach the end, they quickly move down and back to the left for the start of the next line. You repeat the process for subsequent lines, and keep doing this until you reach the bottom of the page. This is essentially the same path a raster scan will follow to draw the lines of a TV frame.
Since the electron gun itself doesn't move, how do we control the position of the beam? Magnets. One set for horizontal movement, and another for vertical movement. Since magnetism and electricity are closely connected, varying the magnetic power for each direction will cause the electrically-charged beam to be diverted and hit a different part of the screen. So, by varying the power to the magnets quickly enough, the beam can be moved in a raster pattern.
Now, if you hold a strong enough magnet near the TV, this will also have an effect on the path of the electrons, and the beam will hit the screen in a slightly different position; thus, the picture will move.
In the case of a colour TV, things are slightly more complex. There are three electron guns (one for each of the primary colours, Red, Green and Blue), and three different colours of phosphor. There is also a metal shadow mask behind the screen, which ensures that the 'red' beam only hits the red phosphors; ditto the green and blue beams and phosphors.
When you bring another magnet close to a colour display, not only are the beams diverted to a different part of the screen, they are also more likely to hit to the wrong colour of phosphor; instead of hitting the red phosphors, the red beam may end up hitting the adjacent blue phosphors, and so on. So, in addition to the picture moving, the colours come out wrong.
Worst of all, the shadow mask is made of metal, and may become permanently magnetised, so that even when you take your magnet away, the nasty effects remain.
If the remaining magnetism is only slight, your TV or monitor's built-in degauss circuit3 may remove it. If not, you may have to degauss manually.4
Moral; don't let magnets anywhere near your monitor- not even speakers, unless they are magnetically-shielded.
Yes, I realise that this writeup is very simplified. It's not intended as a definitive guide to CRT TVs and monitors- which would probably require pictures to explain well.
1The electron gun is a heated cathode; the screen is the anode.
2Strictly, that's 50 or 60 half-screens per second because TV displays are interlaced; but that's not important here. Most monitors display more frames per second, and aren't interlaced, but the basic principle is the same.
3The degauss circuit is the reason some TVs and monitors make a 'grunt' sound whenever they are switched on.
4If the original magnetic source was very strong, you may even have physically distorted the shadow mask. I strongly doubt a fix would be possible in that case.