First, fix your dependencies. To do this, open a terminal and enter this command:
sudo apt-get install -f
After that, update your software:
sudo apt-get update && sudo apt-get dist-upgrade
Don't use the application
Ubuntu Software. The scientific term for it is "complete and utter garbage with tons of bugs which doesn't work 90% of the time and shouldn't be used by anyone for any purpose whatsoever". If you want to use a graphical interface, use
Ubuntu Software Center. You can install it with this command:
sudo apt-get install software-center
It's a bit slower but it works and gets the job done.
If you want to install a Debian package, you can either open it with
Ubuntu Software Center or you can install it on the command line (assuming the package is in your downloads folder):
sudo apt-get install ~/Downloads/thepackage.deb
Note that you probably can install the Debian package directly from the repos without first downloading it. If your Debian package is called
Awesome-Application_1.37_amd64.deb, try this command:
sudo apt-get install awesome-application
It either finds something with that name or tells you that it can't find it. In the latter case, execute the command mentioned two above to install the package you downloaded.