First - which version of Ubuntu are you running?
Next, close Software Centre if it is open and open a terminal. Ctrl-Alt-T is the keyboard shortcut, but it's easily found in the dash. Run these two commands in the terminal:
After the first command, you will be prompted for your login password. You won't see anything echoed to the screen when you type it in, which is disconcerting if you've not experienced this before, but just keep on typing. It is going in. After the second command, you will either be prompted with a "Do you want to continue [Y/n]?" message, or a dependency error message similar to what you got with Software Centre. If the former, just type y and enter and skype will install. If the latter, highlight the whole of the error message with the mouse, right-click -> copy and then paste the error message into your post.
sudo apt-get update
sudo apt-get install skype