I have a red wire along with the black, the white and the copper ground. All four wires were being used in the previous fixture, but I don't know where the red went. The previous fixture had an external plug-in, could it have gone to that? What do I do with it now, just cap it off? Because there is no red wire to connect it to on the new fixture, nor does it have an external plug-in.

One of the two "colored" wires (black and red) is a hot and the other a switched hot.

But either could be either.

You can either try one and wire nut off the other one and see if it works. If not try the other one.

Or you can open up the switch box and see what is wired to what.

The one wire was for the unswitched plug and the other was the switched light.
Cap the black and connect the new fixture to the white and red. If the switch doesn't work then cap the red and connect the fixture to the black and white.