A branch that wasn't supposed to be cut off was cut off. I no longer have that branch. What can I do to make another branch grow back where the other one was?

Nothing, but wait... odds are tree will decide to grow one if there is a bare spot eventually. And it won't be exactly where the old one was...that would be impossible....but it will be close.

If it is a deciduous tree, it may grow another branch close by. If it is a conifer, doubt it. If this branch is/was big, I think the tree will try a wound repair without growth. Also depends where on the trunk. If it was the bottom branch, probably not.