Just to clarify, upload is when you move something from your device to somewhere else; download is when you move something from somewhere else to your device. I'll try to answer both ways:
To upload, press the Share icon within the app your using, and then select Dropbox.
To download, long press the file in question within the Dropbox app, then select Export. Choose the export location (e.g., sd card). This action creates a copy (i.e., the original file will still be in Dropbox unless you explicitly delete it).