How do I determine if a drive is compatible?

Your MacBook Air uses a 1.8 Inch HD.

The only wrinkle here is Apple has used two different HD's one with a ZIF ribbon connector and a CF pin type header connector. You could use the serial number to figure out which it had when you got it new. But if you have bought your system used or it was fixed before you might have a different type of HD than what you expect. I would strongly review the iFixIt guide and then open up your Air to double check which type of HD you have. iFixIt MacBook Air HD replacement guide

As to finding a replacement right now stocks are limited do to the flooding in Thailand for platter type HDs, better yet why don't you look into using a solid state drive (cost a bit more) but offers better reliability.

The problem with airs is prepping the HD and installing software on a fresh drive (if you don't have the Apple Supplied thumb drive with OS-X on it). Hopefully you have a desktop mac or can access someones. What you'll need is an external HD case to hold your new HD or SSD to connect to this Mac to partition, format & install OS-X. Ideally you should get a Firewire case that can hold your 1.8 drive. Don't forget you'll need a cable, either a 400 or 800 (different connectors) depending on what your needs are between the case and the mac your using to prep the drive.