What is the small blue wire in my 240V air conditioner cord?
That blue wire is likely testing the integrity of the ground and if not connected correctly, will not allow it to work. ACs like this create static electricity and that needs to go safely to ground. The 240V will work fine without it, so that blue wire is making sure that you have a good ground connection before it allows it to run.

Along those lines, when you say you tested it with a multimeter, what did you test? The fact that you seemed to think there was a "neutral" on a 240V circuit kind of indicates you don't know what to look for. You need to have 120V from either hot pin to ground, and 240V from hot to hot. There is no neutral. So if you measured no difference from one of the hots to ground, thinking that was neutral, that's your problem. One of your hot's is open. That would explain why your AC is not working.