Which external CD drive is compatible with Macbook Air early 2014?

Apple had a portable USB SuperDrive that they sold for MacBook Airs like yours – but a couple of weeks ago, the online Apple Store listed it as out of stock, and now there's not even a listing. R.I.P. SuperDrive.

Other World Computing has this desktop drive, and a similar desktop CD/DVD/Blu-Ray drive.

Other World Computing – 24X OWC Mercury Pro USB 3.2 (5Gb/s) Super-Multi DVD/CD Burner/Reader External Optical Drive with M-DISC Support

The USB 3.0 B to USB-A cable that comes with it is extremely short, but it's not permanently attached. You can replace either the cable or the power supply without having to replace the drive unit itself.

Most external computer CD, CD/DVD, or CD/DVD/**-Ray drives with a USB interface should be. Some portable drives may want you to plug two USB cables into your computer, so they can draw more electrical power than a single port can provide – so you might want to go for a desktop drive with its own power supply, if you can.

The bigger issue would be software compatibility. I assume that your Mac can

• Play CD-Audio music discs
• Play DVD-Video movie discs
• Burn CD-Audio discs of purchased music that you own
• Read and write CD and DVD data discs

without extra software.

However, if you want to

• Create home video discs using DVD-Video format
• Play Blu-Ray movie discs
• Create home video discs using Blu-Ray Video format

you will need extra software, and then the question will be whether you can still get such software for Big Sur (the most recent version of macOS that old MacBook Air can run).

I have an earlier one of these:

"Pioneer External Blu-ray Drive BDR-XD08S USB 3.2 Gen1 (USB Type-C) / 2.0 Slim Portable BD/DVD/CD Writer Silver"

I use with my Mac Studio. However, I have never been able to watch Blu Ray movies on it.

Apple never included a Blu Ray decoder for its DVD player app in order to watch Blu Ray movies.

I think you can use writeable Blu Ray disks however to store data to achieve an optical back up of data.