My garage door opener is plugged in to two outlets...

There's a black cord running from the opener to the ceiling outlet, as well as a thin brown cable running to a brown box thats plugged into the same outlet, but also bolted/screwed in to the outlet.

I would guess this is common, though I'm changing the battery and don't know if I need to unplug both the brown box and the black cord, or just the black? Is there anything else I need to unplug before changing the battery? Thank you for any info.

You're replacing the battery in what? Your garage door remote? If that's the case then you don't need to unplug anything. You will be fine by just replacing the battery without unplugging anything.

The brown wire/box is the radio receiver which is wired into the operator.