It's most likely an over-heating problem and is shutting down to save the proccessor, common problem with lap tops.
Is it a Toshiba by chance.

To test for over heat problem elevate the laptop so air can flow under it, try placing a fan near it if it runs you know you are dealing with an over heat condition, should take it in for service as over heating will damage the processor.

Thank you for your help. I have done as you suggested, and it still shuts off. I am thinking that the over heating may not be the problem.
It is a DELL Inspiron 1000.
I have let it set over night, so it would be completely cool, tried starting it up again this morning, thinking it may be the heating problem, but it did the same thing.
Thank you for your help....
Any other suggestions?
I really appreciate it.

Yes, letting it sit over nite really does not matter, they heat up instantly, Does it feel hot? The other possibility is it is a (BSOD) Do you see a blue screen right before it shuts down? They can occur very quickly & you may miss it. it is a blue screen with an error on it, If that is the case try this setting after you do you will now be able to see blue screen error, we would need this error to help you, here are the steps

1.
Right-click My Computer, and then click Properties.

Click the Advanced tab.

Under Startup and Recovery, click Settings to open the Startup and Recovery dialog box.

Clear the Automatically restart check box, and click OK the necessary number of times.

Restart your computer for the settings to take effect.

Now when you go to shut down and a fatal error occurs, you'll at least see it and it won't cause an automatic reboot. You still have to sort out what's causing the problem.

You never mentioned if you use the computer on your bed or at a desk. If on a soft surface, the fan vent holes will be blocked which will cause overheating. If it does this on a desk, the fans have undoubtedly drawn in animal/human hair, dust etc and clogged up the cpu which causes overheating. Best bet is to have it serviced/cleaned as you have to tear the laptop apart to get to the cpu

Might open a 'Computer Managment' window and look in Device Manager & Event Viewer for errors. Maybe you'll get lucky and see what's up.

fwiw

My laptop ac adapters, I use two in seperate locations, where they plug into the computer have been yanked out so many times by tripping over them that it is a flaky connection at times. I have to straighten the cord ends just right for it to not shut down like you've described - works fine until I trip over it again and then starts the sudden instant off's with a minor movement of the cord.

I've also had a screen switch with a flaky connection. You know.. Close the lid and shuts off... It corroded or maybe had too many crumbs in it. By pushing it on/off with a finger several times it seemed to munch up those crumbs which cleaned its contacts. It was causing funny shut downs with a bad connection.

i use the computer at a desk. I regularly blow out the computer including the vent with air duster. I have the laptop sitting on en external fan that I purchased to prevent overheating.
It's not the battery because I keep the computer plugged into the wall since my battery is bad.
The cord is still in very good shape.

Check out this link for a solution that others have found works for them.

Here is a link that might be useful: Inspiron 1000 overheating