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First, fix your dependencies. To do this, open a terminal and enter this command:

sudo apt-get install -f
sudo apt autoremove

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.