I just purchased Baldur's Gate, down loaded it, and it won't work.

A little more information could be helpful. :) What is your operating system? What do you exactly mean by won't work? Does the game crash to desktop on startup? Do you get any type of error message?

Says a problem occured that caused this program to stop responding with windows.

Are you using EE or original? If you playing the original, try some of the compatibility options. I usually go with WIndows XP service pack 3 when using compatibility mode.

I'm not sure what the difference is

If you have the Enhanced Edition, it will say so beside the title. Try compatibility mode, and if it doesn't work, try opening a ticket for support. Your anti-virus or firewall might also be blocking the game. So check to see if your program allows Baldur's Gate to run. I'd recommend compatibility mode first though. If compatibility mode doesn't work, remember to restore it to its original settings before trying other options.

Edit: I forgot to mention that the pic I enclosed is what your list might look like when you open your start menu. Sorry about that.

I'm not as computer savyy as I used to be. How do I get to compatibility mode??

Right click on your desktop icon and a box will pop up. Click on "properties". Another box will pop up. There would be a tab called "compatibility", click that. There will be a box, with the text beside that saying "Run this program in compatibility mode for:" Make sure that is checked. Click the drop down menu, and go to "Windows XP (Service Pack 3)". At the bottom of the box, click "Apply", then click "OK". After that, try starting up the game.

Not seeing the "compatibility" tab

I
have a problem running all the "enhanced" games. Running windows 7, after install, I try to play, and get a black screen, but can hear game audio in background.

Go to the folder where you have the game installed. Find the exe file which runs the game. Try doing the above with it.
Same here. Try to use the compatibility mode. Also make sure you run the game as an administrator and that you verified the game data through GOG Galaxy (you don't have to use it to play though).

I'm not seeing the exe file

I got the game working. Went into desktop, found the program file for GOG, found the file for Baldur's Gate and clicked on Launch Baldur's Gate and it worked

Now I'm having a new problem. I keep getting booted out of the game every few minutes.

Sounds like antivirus.