My wife drives a 2008 Honda Civic with about 47,000 miles on it, and today when I was riding with her I noticed that when the AC is on a chirp sound comes from the engine compartment. We parked the car but left the enging running with the AC off, no chirp. As soon as we turned the AC on the chrip started with us parked but engine on. We turned the AC back off and the chirp stopped at the same speed the air stopped coming out of the vents. (in other words, it wasn’t just all of a sudden gone, the sound took about 2 seconds to go away and as it did it got quieter and the speed of the chirp slowed down.
Any indication from this what the problem is? Is it definitly an air compressor or could it just be a belt? Looking for some guidance and what you think it will cost to repair this.
Also, I am NOT a mechanical guy at all, I do computer work so I doubt I could tackle this myself. When I showed my wife the diagram of the AC system she said “stop right there” so she would much rather spend a bit more to get it fixed right then have no AC at all when I get done with it. 
The belt needs to be replaced.
Your description seems to point to the blower motor to me. Try turning the fan on and off, with the AC compressor off. See if you get the noise then. I would agree with the bad/loose belt suggestion, but for your statement of "We turned the AC back off and the chirp stopped at the same speed the air stopped coming out of the vents. (in other words, it wasn’t just all of a sudden gone, the sound took about 2 seconds to go away and as it did it got quieter and the speed of the chirp slowed down. "