I've never seen a good explanation about how to hook up an AC motor. All the explanations are different and they all over-complicate the matter. There are 2 coils of copper wire and they both get hooked up to the AC current. You add a capacitor to in series with one coil to get the motor to turn. Hot and common don't matter - you're hooking them together with a long coil of wire.
Well fear not! All you need to figure it out is the face plate, the appropriate capacitor, and a cheap multimeter. Using this method, you should be able to wire up any single phase 100/120 VAC 50/60 Hz motor regardless of the wiring configuration. The face plate will tell you what size capacitor you need.
If there are two capacitors required, the face plate will list both. Also there is often a great big hump where the capacitors get mounted. In this case, the capacitor wires will likely be routed to the capacitors already. If the capacitor is already hooked up, you may not be able to find both sides of both coils. One of the coils will already be hooked to the capacitor in series and will have a high resistance. You can verify that one side of the capacitor is attached to a coil of you have access to the leads.
If you're lucky, there are 3 leads. In this case, all three wires are connected to each other with different resistances between them. Find the pair with the highest resistance between them and connect those two wires to the capacitor. The third wire goes to one leg of the AC (either hot or common it doesn't matter) The other leg of the AC should go on one side of the capacitor. That's it for the 3 wire setup. Switch the side of the capacitor to reverse directions.
With four wires, you may have access to both sides of each coil. You should have two pairs of wires each attached by a coil with a corresponding resistance. In my case, with a tiny 1/50 HP motor, I had one coil which measured 70 ohms and one was 120 ohms. Tie one side of each pair together and connect that to the AC. Tie the other two ends to the capacitor. Attach the other leg of the AC to either side of the capacitor.
Another possibility with four wires is that two wires will be tied together with almost no resistance. In this case (ignoring one of the pair for the moment) you have a three wire setup. The second wire is for temporarily attaching a second start capacitor. This start capacitor typically goes in parallel with the the other capacitor to increase the starting torque. Some, especially larger, motors will have both a start capacitor and a run capacitor. The start capacitor can be removed manually, or may have a speed switch that automatically disconnects it.
5 OR MORE
If you have 5 or more wires, then likely one or more pairs are connected internally or you don have extra built-in accessories such as a clutch or brake. There should be some mention of extra parts on the face plate. Also, the clutch or brake are usually removable or at the very least are manufactured separately and should be obvious. The motor will look too long. You should still be able to identify the two coils with a volt meter, tie two ends together, and tie the other two ends to the capacitor.
A good final check is to measure the resistance across the power leads BEFORE YOU PLUG IT IN. If you have zero resistance, don't plug it in. Bad things happen and typically the lights go out after some sparks fly... Your two coils are in parallel now and resistors in parallel add according to the rule:
You can reverse the direction by switching the side of the capacitor that gets a power lead. (Swapping the power leads with each other doesn't do anything.) You shouldn't try to switch directions with the motor spinning. Again, this typically leads to spark, but not from the wall. You'll likely fry whatever switch you're using to switch directions. I learned that one the hard way.
Go forth and spin AC motors! Find the end of each coil with a multimeter, connect one side of each coil to the power and the other side to the capacitor. Connect the other side of the AC to one side of the capacitor. Hot and common don't matter at all.
Don't work on the motor with the power connected! Be careful with the capacitor. A small one probably won't kill you if you accidentally discharge it, but they hurt. Check the resistance between the AC leads before you plug it in. Shorting the AC is a bad thing.