What package are you trying to install? Say you were trying to install the Chromium web browser. Open a terminal with Ctrl+Alt+t and type
when asked for your password enter it and then copy and paste the output on this form. Note: use any package you want as long as you know the right name.
sudo apt-get install chromium-browser
If it installs with no errors then we know that the package management software is working and its only the software center itself.
Edit: I did not notice that you were not using Ubuntu. The keyboard shortcut to get to the terminal may not work but you should be able to open up the terminal from the menu.