How to fix broken packages
'Broken packages' are packages that have unsatisfied dependencies. If broken packages are detected, Synaptic will not allow any further changes to the system until all broken packages have been fixed.
To fix broken packages
Choose Edit > Fix Broken Packages from the menu.
Choose Apply Marked Changes from the Edit menu or press Ctrl + P.
Confirm the summary of changes and click Apply.
If that does not help, then please follow this procedure:
How to free disk space
Settings -> Preferences -> Files -> Delete downloaded packages after installation then click Delete Cached Package Files.
Broken Upgrade or Installation
What to do if an installation process fails and you find it is no longer possible to install or remove packages:
Open a Terminal and type the following commands, pressing the Return or Enter key after each (you may have to type in your password):
sudo dpkg --configure -a
sudo apt-get install -f
Browser works, Synaptic will not
(note: probably inaccurate or outdated)
If the router is set to auto discover DNS then Synaptic will fail. Resolve this by setting up a static ip address and manually set DNS ip in router. Now Ubuntu (system-administration-networking) config static ip and enter your ISP dns ip and delete router DNS ip which will look something like 192.168.1.1