My Compaq keyboard has a key with an ico...

The key with the moon on it will put the computer to sleep. It shuts down, but is not "turned off" This means that it will restart quickly.
The light with the arrow pointing down shows you that the shift lock is on.
You probably have power saving turned on. I am assuming you are running XP. Right click on the desktop. Select properties then click on the screen saver tab. at the bottom of the screen there is a button for the power saver. click on it and increase the screen blackout time (or set it to never)

The Moon key is for putting your computer to sleep, the keypad light is probably the NUM LOCK LIGHT. When your screen blacks out will it come on again if you press enter on click the mouse button? if so, go into the control panel, click on the power options and look for the icon marked 'Screen' or something similiar. Change the timing to an hour or so...