There is a Dropbox app for Android, but it does not work like the Dropbox apps on Windows or Linux. It does not sync your Dropbox directory to a local directory. Instead, it only allows you to access files in your Dropbox directory through the app.
If you prefer to have your Dropbox directory synced to a local directory on your Android device, then you might want to try the Dropsync app. It supports both two-way sync and one-way sync. The free app only syncs a single directory inside your Dropbox directory and all the files and subdirectories inside it. If you need to a more fine-grained sync of multiple directories and exclusion of certain subdirectories from sync, you will need to get the Pro app, which is a paid version.
Tried with: Android 4.4.2 and Moto G