You can format the drive with Disk Utility, as described here: Erase and reformat a storage device in Disk Utility on Mac - Apple Support
Most drives will come with the wrong file system format, even if the box is claiming "formatted for Mac". This just means, that a Mac can read files on that drive, but does not mean that it is save to store a library there.
If you are getting yourself a new external drive, buy one with enough storage to last for years. The drive should have enough storage to hold at least twice the size of your Photos Library, better even more, because repairing or upgrading the Photos Library will need plenty of extra storage. occasionally you will need to make a duplicate copy of the library, before start with major changes in your library, like weeding out duplicates or similar. then you will need enough storage to create a duplicate on the same volume as a current safte copy.
For a MacBook Air I would get a small portable drive, so it will not be a hardship to keep it connected to the MacBook Air at all times. A drive with your Photos Library needs to be plugged into the Mac, whenever you are signed into your user account, even if you are currently not working with Photos, or the faces scan and other background processes for Photos will never finish.