Lightroom is designed as a single-user, single-computer program. To sync two computers requires some hoops to jump through.
Lightroom Mobile will sync one computer to multiple mobile devices.
The easiest (not prettiest) method of syncing multiple computers is to put the images and catalog on an external drive and move the drive from computer to computer as needed.
Sync can only be possible in between Lightroom Mobile and Lightroom Desktop application, therefor if you have iPhone Or iPad you can install the Lightroom Mobile app and sync your collection either from Desktop or notebook.