1. You haven't described your download type. Is it regular (http/ftp) download, download from file hosting, flash video, torrent or anything else? Some download types cannot be resumed after stopping, some cannot be even restarted from the beginning.
2. Open main FDM window, select "Downloads" tab, go to the lower pane, select "Log" tab and read, what is wrong with your download. If the log is empty, go to FDM menu Options | Settings | Downloads | Essential and make sure "Detailed log" is turned on, then start your download and observe, what goes wrong.
3. Downloads that can be resumed are displayed in green. Right click the download name and make sure "Start download automatically" is marked/checked in the context menu. For a new download make sure "Start automatically" is selected in "Add download" dialog window. For all new downloads go to FDM menu Options | Settings | Downloads | New download nad make sure "Start download automatically" is marked.