It appears that you may be stuck because of a recent application update. I will give you some steps to try. Please note that it is very important to follow these steps specifically in order, as failure to do so may result in other issues:
1) Click on the Dropbox icon in the system tray and then click your profile picture or initials
2) Select 'Preferences'
3) Click on the 'Account' tab, and then click in the window under the tab (just anywhere will do)
4) Hold down the 'alt+H' keys on your keyboard
5) Click on the 'Fix Hardlinks' button
6) You may be prompted for your a username and password, this is your computer login credentials, not your Dropbox account ones
7) Please wait about ten minutes to see if Dropbox resumes syncing.
If the above steps don’t resolve your issue, please repeat steps 1→5, but at step 6, this time click on the “Fix Permissions” button.
Please note that if you use “Fix Permissions” syncing will start back over. If you need to use this step, please allow for a few hours to pass and let me know if syncing becomes stuck again.