Here's what you should do:
1. Check to see if the app you want is in your distribution's repositories. Most Linux distros don't follow the Windows paradigm of software installation (download from developer). Instead, they maintain a central repository and let you install it with a package manager. What distribution do you use?
2. If it isn't in the repos, you may need to either compile it or use a binary install script. A tar.gz file is nothing more than an archive to be extracted. You can extract it with a graphical tool or by entering
tar xvzf filename.tar.gz
in a shell.
3. If it contains either a .sh or .bin file, you can run the installer with
chmod +x filename.sh
4. If it contains a bunch of files with names like "configure" or "Makefile", it needs to be compiled. You can usually do this with the following, though there are exceptions:
sudo make install (if you use a distro with sudo)
What are you trying to install?