how can I transfer my photo library from my Macbook air on the desktop?

You can use Migration Assistant (Included on every Mac) Applications>Utilities>Migration Assistant. Otherwise a Flash Drive also works.

Cheers,

Adam

If your not into using iCloud, then use your local network.

see > OS X: Manually migrating data from another Mac - Apple Support

and/or > http://www.techradar.com/us/news/networking/how-to-share-files-over-a-network-in -os-x