You've probably downloaded a source tarball. Usually what that means is that you have to install libraries/utilities that are required to the compilation of your software and then configure, build and install the compiled program.
There's no single way to do that, usually when you unpack your tarball, there's either a README* or INSTALL* file that describes what needs to be done to install it. It's fairly common to just:
There's a drawback to this method - you'll end up with files installed completely outside of your package management system that your PMS has no idea about and cannot manage. It's usually fine for temporary testing and if you know what you're doing and keep files installed in such a way in a separate directory.
You said you're not used to working in the terminal - unfortunately working in the terminal is the least complicated part of compiling programs from scratch.
You might want to find your application among packages already available to Ubuntu and download the Ubuntu distribution on the DVD and then use apt-cdrom to add it as a source of available packages.
PS. if what you've downloaded is an already built application, you can view the contents using:
tar tjf program.tar.bz2
tar xjf program.tar.bz2
but you have to be sure that you know what you're running. I'd look for the way to run your application in one of the files within that archive.