NSProgress` is an object in Foundation that represents the completion of some work, allowing easy reporting of progress across various components for both UI and system applications. Couchbase Mobile uses `NSProgress` to inform about change events during replication. The tutorial on importing data from the Google Places API to Sync Gateway and replicating it to an iOS app demonstrates how to use `NSProgress`. The process involves creating a new project in the Google Developer Console, generating a Server API Key, setting up a pull replication with the iOS SDK, and using replication change notifications to display a progress bar in the UI.