What do you mean "The new light is not a dimmer."?
If it cannot be dimmed, you need a different switch.

If it can be dimmed:
White fixture wire to the box whites;
Black fixture wire to the box red;
Bare fixture wire to the box grounds (bare or green).
Do not attach anything to the box blacks.

You can bet the existing wiring needs to be connected as before. If left disconnected, something, or everything, on that circuit will no longer function.

(OLD HOUSE) - Original wiring had a dimmer switch which controlled 2 separate ceiling lights (2 bulbs each). The switch was changed to a standard light-switch and one ceiling light was changed to a ceiling fan with lights. Now, we are attempting to change the other ceiling light to a new ceiling light with 3 bulbs. The ceiling wiring is different in that it has 1 black, and 4 off-white wires. The new fixture has 3 black & 3 white wires. We've connected the 3 new black to black & 3 new white to the white, attaching with proper connectors. What happens with the 3 extra white wires?

You should have started a new thread, rather than posting on an old one. The title of the thread is confusing the issue.

What happens with the 3 extra white wires?

The go exactly where they were before you took the old fixture down. You DID make a drawing or take a photo, I hope.