When taking a picture the mirror inside the camera stays up untill you remove the lens

This is a common problem that these cameras came with, its a "glitch" in the motherboard that "turns on" a chip inside of it that pauses or freezes the normal function of the camera and it is triggered by any function/s that use that specific chip. Best thing you can do is to either get another board and install it yourself (if the camera is out of warranty) if not, don't mess with it and send it to Canon for a replacement (not a repair) of the motherboard.

Since it's a film camera, there are a couple of things to check:
1. You have left the lens cap on.
2. If this is a single lens reflex camera, take the lens off and make sure that the mirror isn't stuck in the "up" position. If it is, GENTLY pull it down until it stays down. At this point, your shutter may be open. Try cocking the shutter again, fire off a test shot, and see if the mirror sticks in the up position. If it does, you probably need to replace the mirror bumper foam - age can make foam deteriorate into a gooey, tar-like substance that can make the mirror stick in the up position.
3. Does the camera have a T (for Time) setting on the shutter speed dial? If so, turn it off the Time setting and set it to a manual shutter speed such as 1/60 - or whatever your 'X' synch speed is.
4. Your batteries died in mid-exposure. Replace the batteries. You may have enough juice in them to show the viewfinder display, but not enough to finish the exposure and close the shutter.

If the camera appears to be working properly in some pictures and not others I'm going to suggest you look at your Shooting Modes. It's very possible that the mode dial has been turned to the TV mode which is Shutter Priority. With the Rebel having a shutter speed of 30 seconds to 1/4000 you may click the shutter to take a pictures and the camera appear not to take the picture. A few seconds later the shutter closes and the mirror drops. This also can happen if you are in the Night Portrait mode. The length of time my be longer if the lens cap is immediately placed on the lens.

Change your mode to "P" program until you become more familiar with the camera functions and settings

Mirror lockup is probably enabled. You can disable it by going into Menu and then Custom Functions. Select Mirror lockup and disable it.

Try cleaning the contacts on the lens and camera body where they meet with a cotton swab lightly dampened with rubbing alcohol.

Try cleaning the contacts on the lens and camera body where they meet with a cotton swab lightly dampened with rubbing alcohol.

The view finder "going black" on an SLR is caused by the mirror being up, which is also when the camera is actually taking the picture! The mirror sends the light from the lens to either the viewfinder or the sensor. When it's down, the light goes to the viewfinder and you can compose the picture. When you press the shutter release, the mirror comes up, and light goes to the sensor. In a film SLR, there was a shutter just in front of the film that controlled the amount of light that hit the film. In your D80, it's a combination of the mirror and the electronics that set "shutter speed".

What is happening when you get these white pictures is that somehow your settings are holding the mirror open far longer than it should be open. When you turn off the camera, it drops the mirror automatically. If you are in S or M, it's because you've set your shutter speed improperly. If it's doing this in Auto mode, then there may be a problem with the exposure meter or the lens.

If you describe more about when it happens (shutter speed settings, Auto or manual mode) I can probably tell you if it's your settings or the camera/lens.

Do you mean that, you press the shutter release once and the shutter stays open until you press the shutter release a second time. If so, check your shutter speed setting. If it is set to "T" (Timed Exposure) the camera is working correctly. Change your shutter speed to an appropriate setting, such as 1/125. If that does not solve the problem, you might need to take it in for repair.

Usually tele lenses need longer exposure,and mirror lenses need more than normal lenses.Try with 1sec, half sec,( 2 in the speed dial, not 2" ) or quarter sec (4 in the speed dial) exposures.

One more thing, you need good sturdy stands to hold cameras.

Check for the obvious -- shutter speed, aperture, and ISO set appropriately to give you a well-exposed image, and lens caps off. If none of those are the problem, then there's a problem with the mirror, the shutter, or the sensor itself. You can carefully look inside the mirror box with the lens removed, and see if anything is blocking the motion of the mirror. If you don't see anything (or, honestly, if you do), take it to a trusted camera shop or call Olympus's repair center.