I have been using Lr standalone for years on 2 PCs but am considering the cloud.
I have no idea how it works but know it can work on mobile devices like smartphone & ipads but not sure about mobile on my laptop.
I want to have my entire catalog/library on my home desktop and laptop. My ? is if I make edits on one computer will they be synched with the other computer? if so how does it work? auto synch?
Lr CC works exactly as does your current version of Lightroom. Photos and catalogs are stored locally on you laptop and not in any cloud servers. The cloud component involves syncing to mobile devices (iPhone, iPad, android phones) via Adobe servers. Syncing does not operate across laptop/desktop computers.
See this: Lightroom and Creative Cloud FAQ
Adding to what Wolf said, Lightroom mobile is only used for syncing your images in your collection to other mobile devices. It will not sync Lightroom catalog. To know more, check this Adobe Photoshop Lightroom Help | Syncing Lightroom - Desktop and the Mobile app