I have replaced a broken light fixture which is on a three-way switch. I connected the two neutral wires, and connected the new fixture's two wires to the two hot wires. Now one switch has to be on in order for the second switch to work. What do I need to change for either switch to work independently?

Assuming that you did not make any changes at either switch, tell us what wires you have in the ceiling box (by color). There should not be two neutrals unless you mean two neutral wires to the fixture.

The light box has two red wires (already connected), two neutral wires and two hot wires, and two ground wires. Again, I connected the two neutral wires to each other, and connected the hot and neutral wires of the light fixture to the two hot wires in the light box.

OK un do it. One of the white wires is the neutral. The other white wire is a traveler to the second switch. Do you have a meter to determine which one is the neutral?

Remove the switch plate covers. One switch will have a two wire cable with a black and a white wire and a three wire cable with black, red and white wires. The two white wires will be connected together with a wire nut. The black of the two wire cable will be connected to the common screw of the switch. The red and black wires of the three wire cable will be connected to the traveler screws of the switch. The common screw will be black or a darker color screw.

At the other switch there will only be the three wire cable. We need to know which wire is connected to the common screw. It should be black but may not be.

Hopefully I'll describe this correctly:
On the switch, there are two screws on the left, and only one on the right. The two hot wires are connected to the top screw on the left, and the sole screw on the right, and the red wire is connected to the bottom screw on the left. The two white (neutral) wires are not connected to the switch, but rather are connected to each other. It seems to me that this switch is not meant to operate as a three-way switch. Am I correct?

That is a 3- way switch. Go to the other switch and see what color wire is connected to the common screw. You will have to pull the switch out to see the color of the screw. Not all switch are made the same. Usually the common screw is on the side with two screws. Sometimes on the top and some times on the bottom. After we have insured that the black wire is connected to the common screw we will know how to connect the wires in the ceiling box.

From what I can tell, it is the red wire that is connected to the common screw on both switches.

Not if they are connected together in the ceiling box.

I must be wrong, and that the black wires are, in fact, the ones connected to the common screws, because the two red wire in the light box are connected to each other. The screws on the two switches all seem to be the same color. I am having a hard time determining color differences between them.

In the first box, the one with the 2-wire cable and the 3-wire cable, the black of the 2-wire cable is connected to the common screw.

I agree.

Which screw is the black of the 2-wire cable connected to in the first switch box?

The common screw will be a darker color such as black or brass. The traveler screws should be lighter color or silver.

The top screw on the left, while the red wire of the three-wire cable is connected to the bottomw screw on that side, and the black wire of the three-wire cable is connected to the lone screw on the right side.

So top screw on left is common. On the second switch, what color wire is connected to top screw on left?

Black, while the red wire is on the bottom screw on the left, and the white wire is connected to the lone screw on the left.

I meant to say lone screw on the reight side. Sorry.

Good. Now we have to determine which one of the white wires in the ceiling box is neutral and which is the traveler. Turn power off. Connect the white of the light fixture to one of the whites in the ceiling box. Connect the black of the light to the black in the same cable. Put wire nuts on the other black and white or make sure they are not touching anything.
Turn power on. If light burns you are connected to neutral. If light does not burn flip the first switch. If light burns you are connected to neutral. If light does not burn with switch in either position, the other white is the neutral.

Connect the white of light to neutral. Connect black of that same cable to white of the other cable. Connect the black of the other cable to the black of the light.

Thank you, Harold.

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