I want to display all files and folders containing in Dropbox. Also user should be able to view, update and delete files or folders of dropbox.
for this which Dropbox API I need to use?
whether it should be Core API or Sync API?
For Android, and the functionality listed, the Sync API is recommended: https://www.dropbox.com/developers/sync
The Sync API is easier to implement and does a lot of the work for you. However, note that it doesn't support Full Dropbox access:
https://www.dropbox.com/developers/reference/devguide#app-permissions
So, if you need Full Dropbox access, or any other functionality/platform support not provided by the Sync API, then you'll need to use the Core API