Which backup drive to purchase for MacBook Air (M4 chip)?

Usual would be a hard disk drive with probably 2.5 to 3 time the internal storage of the Mac, and usually with a USB-C connection for ease of use.

An external SSD will also work and will be faster, but HDD speeds matter less here and the higher capacity of an HDD (for a given price) is usually the better trade-off.

If the budget is available for this, a NAS with Time Machine server support avoids having to cable the MacBook to the HDD. Backups can happen automatically whenever the MacBook Air is connected to the same Wi-Fi, with no cabling required.

Another Mac with enough storage will also work as a Time Machine target.

How to use Time Machine: https://support.apple.com/en-us/104984

What storage can be used: https://support.apple.com/en-us/102423

I’d have to agree with Bob. I’m using a Samsung T7 and quite happy with it.

I would opt for an external SSD over a HD. My preference is based on speed and reliability.

If this was a year ago, I would have suggested a 2 TB Crucial X9 SSD (which I use for TM backups on my M4 Mac Mini Pro, 2020 27-in iMac, and M1 MacBook Pro M1 Pro. However, as of now, Crucial has exited the consumer business market, so a portable alternative might be found at OWC (macsales dot com) and dependent upon your budget.