As the title says, I have an older computer and I want to move up to a quad core. The list on the manufacturer's site is pretty incomplete with the number of AM3+ processors being sold.
The board in question is Gigabye GA-880GM-D2H (rev. 3.1)
No, not every processor will work in every board, only the supported processors for that board will work.
You can find a full list of supported CPUs for your Motherboard on the official Gigabyte page for the Motherboard.
Here is the direct link for the Supported CPU list:
Retrieving data ...