tar zxvf <filename>
This will create a directory that usually has the same name as the tarball, but without the tar.gz suffix.
Look for a file named README and/or a file named INSTALL
These will usually have instructions on how to install the package. Note that to install on Fedora, you will need at least a few extra programs that aren't installed by default.
yum install gcc gcc-c++ make kernel-devel
(If you have a PAE kernel, use kernel-PAE-devel instead.)
Note that is the bare minimum of what is needed. There is an entire group of packages called Development Tools which will definitely have everything you need.
To install that instead of the minimum mentioned above
yum install @development-tools
yum groupinstall "Development Tools"
(either command will work.)
Most of the time, the README or INSTALL file will tell you to run
su -c 'make install'
(You usually have to be root to install a package)
but sometimes the instructions will differ. Also, most third party tarballs will, by default, install into /usr/local/bin rather than /usr/bin (which is where most Fedora packages are installed.) In practice, this doesn't matter that much but it can be handy to keep tarballs separate from rpm installed programs.
Lastly, are you sure there is no rpm for the program you want? If there is, you're almost always better off using yum, as it will take care of dependencies for you. Many third party tarballs are written for Ubuntu/Debian and will look for dependencies that use Ubuntu or Debian naming conventions.