So you decided to download a program for Ubuntu in apt format.
How do you run to install it? Well you can make it via the tool of choice – black terminal window, but normal people prefer some graphical interface. So when your browser asks to select a program to run this file, go to Filesystem, then Usr, then bin and select software-center. Make sure to select this program as default for APT links. Now clicking the apt link will automatically launch Ubuntu software center, where you can install the program the usual way.