Unlike the Dropbox desktop app, which constantly checks your files for changes, the mobile app usually syncs on demand only. This prevents Dropbox from consuming all of your bandwidth and space.
The mobile app shows you the list of your files, but it only downloads them as needed, such as when you tap on a file to view it. To ensure that a file is always on your device, mark it as a favorite by viewing it in the app and tapping the star icon at the bottom of the screen.
The mobile app can upload photos and videos from your device automatically using the Camera Upload feature. You can also manually upload photos, videos, and other types of files. See "How do I upload files?" for more.
The mobile app tries to be smart about how much syncing it does based on the type of network your device is connected to.