Here is a diagram of the outlet. Hope that helps.

We installed the overhead light, but it only works when there is something plugged into the switched socket.

What cable in your diagram is going to the overhead light? Or is this BEFORE you messed with the wiring?

It looks like cable 1 is a switch loop going to the light switch.

Either cable 2 or 3 is the power source. The other is feeding power downstream.

Is the reason that the overhead light only works when something is plugged-in because the tab was removed from the outlet to make it half hot?

Nope. The removed tab just makes it so the bottom can be switched and the top is constant power. You did something funky and wrong with the wiring to get the behaviour you're describing.

Assuming your diagram above is the BEFORE picture, here's what your AFTER picture should look like.

(Grey boxes are marretts)

1. Do you still wish the half of the outlet to be switched? or neither?

2. Do you wish for the old switch to control the new light and not the outlet? or both the light and the outlet?

3. Do you know if the hot comes from the switch or the outlet? Hint: pull the switch out and see how many wires are in there.

4. what is that word in the little box? I can't read it.

You've connected the ceiling light in parallel with the switch. That puts it in series with the receptacle. That's not right. It's not even wrong. :)

With the switch off, the mains voltage is divided between what's plugged into the receptacle, and the ceiling light.

And it's divided in inverse proportion to the size of the loads. Assuming incandescent bulbs in both, plug a night light into the receptacle, and the night light will light up to nearly full brightness while the ceiling light will barely glow, if at all. Plug an electric heater into the receptacle, and the ceiling light will light up to nearly full brightness, but the heater won't do much.

I suggest that you go to your local public library and check out a book on basic home electrical wiring. Read the entire book, especially any chapters on basic electrical circuit principles. Make sure you read about switch loops. By the time you get done with the book, you'll understand what you did wrong. :)

If you're busy and don't have time for that (I understand!), it's probably better and safer to call in a pro.

PS: Don't feel bad; I did something similar myself once, many many years ago. The house was wired with knob and tube. I was trying to install a new receptacle in the open-floor attic. (This was back when the code still allowed you to extend a K&T circuit with NM.) I thought I was connecting to wires that fed a receptacle in the wall of the room below, but I actually connected to a switch loop one stud bay over. Hilarity ensued, sort of.