Mac storage basics · Photos, music and video
What is the System Photo Library in Photos?
Short answer: It's the one Photos library used for iCloud Photos and sharing with other apps. Set it in Photos → Settings → General with Use as System Photo Library.
The System Photo Library is the one library Photos uses for iCloud Photos, Shared Albums and for sharing pictures with other apps. You can have other libraries, but only the system one syncs with iCloud.
To change it, open the library you want in Photos (hold Option at launch to choose), then go to Photos → Settings → General and click Use as System Photo Library. Switching can trigger a long iCloud sync, so do it deliberately.
For moving the library to an external drive with iCloud in mind, see iCloud & iCloud Photos.