The first ati-driver-installer etc. looks like a binary installer. first you need to make it executable
chmod +x ati-driver-installer-8.24.8-run
that should work for that one which I'm assuming is the ati one (the one i'd prefer)
the second is in an archive called a tarball. to keep this simple you could probably just use a graphical program , but if you want to do it in the terminal it is
tar xzf fglrx-driver-8.24.8.orig.tar.gz
once you have done that go into the directory and depending on the software you are trying to install
apt-get offers a way of downloading and installing things with resolved dependencies with a single line at the command line. It is the easiest way to get new software if it is available and your repositories are set up.
Installing software nearly always requires superuser credentials, su is sufficient. You shouldn't need to log in as root at all. The only time I do is if I have a lot of things to do at once.