Offline mode is, quite simply, when your phone does not have internet access. Unlike the desktop web app, on the mobile apps you can also work without an internet connection.
This is extremely useful for working while you're on the move - Whether you're in the sky, where you might not have access to internet but you still want to take some notes before your latest meeting in a far flung place, or on the train on the way to the office.
How does it work?
It's quite simple really - when you're offline, your phone will pick up from the last synced data. You can keep working on this data, moving deals along your pipeline, adding notes, activities, and anything you would normally do, then when your phone is back online Pipedrive will sync as usual and your data will be matched on the web app. Simple!