How do I move photo library from iCloud to a folder on MacBook Air?

Open Finder on your Mac, use the Finder "Go" menu and click iCloud Drive. Then find the photos library file and move it to another folder. The usual location for it is the "Pictures" folder in your Home folder (Finder "Go" menu > Home). You can open two Finder windows on your Mac and with the iCloud Drive folder in one and the Pictures folder in the other, drag the file from one window to the other.

If you don't have iCloud Drive turned on in your Mac there won't be an iCloud Drive item in the Finder "Go" menu. IN that case you can use an internet browser, log into www.icloud.com and go to iCloud Drive there and download the photos library file and move it from Downloads to the Pictures folder.

After you have moved the file, you may need to hold down the option key while opening the Photos app on your Mac and then select the photos library file in its new location.

After all of the above, if you want your Photos to be synced with iCloud, turn on iCloud Photos instead of moving the photos library to iCloud Drive. See: Set up and use iCloud Photos - Apple Support

Are you saying that you are using iCloud Photos on your iPhone, but not on your Mac? If so, then you can turn on iCloud Photos on your Mac, for the same Apple ID, and your iPhone and Mac will synchronize against iCloud (and indirectly, against each other).

If you are trying to copy photos out of the iCloud Photos system into "loose" files on your Mac that are separate from it, you can use the File > Export menu item in Photos. Note that if a photo has multiple associated versions (e.g., original and edited), you might only be able to export one at a time.

Another possibility is to set up your Mac to download full-size originals of all photos in iCloud Photos to your main Photos Library. Then once it is caught up, you could make a copy of that entire Photos library including all of the full-size local copies of photos stored inside it. Macs can have multiple Photos libraries (though you can only use one at a time, and at most one can be synced to iCloud Photos), so you could presumably use the copy as a local library.

I've seen a number of posts where people have put the photos library file in the iCloud Drive folder on a Mac. When they do that it and then open the Photos app it results in an error message saying that the photos library needs to be moved. I think that is what the original post is about.

I don't see a Photos Library file on my iCloud Drive – and I'm using iCloud Photos.

I'm not using the feature to store the Desktop and Documents folders in iCloud, if that makes a difference.

[EDIT:]. Oh, I see what you're saying. Some people manually put their Photos library file into iCloud Drive – and were hoping they could get away with having the whole thing on the equivalent of a NAS – rather than using iCloud Photos.

Right. It seems to be fairly common and the Photos app (at least in the last year or so) detects that the library is in that location and gives an error message rather than opening it.