There are 2 problems you might be seeing here:
1.) One problem is known as visual 'tearing', which happens when the framerate is faster than the screen refresh rate. Its extremly noticible in fast moving games, and there is an obvious 'tear' horizontally across the screen where one half has not quite caught up with the 2nd half.
The fix for this is easy - enable "Vsync" in the game you're playing. This will cap your framerate to your monitors refresh rate, and completly eliminate the tearing
2.) The other common problem with LCD's is ghosting, where the pixels in the monitors can't change fast enough, which causes a "ghost" or a "streak" image to appear in fast moving games or movies. This is quite common with LCD's, and CRT's are not affected. The only solution to this problem is to get an LCD with a faster pixel response time.