A ".tar.gz" file is a compressed archive. It can contain any sort of data. Assuming it belongs to a program, it is either a "source tarball", or a binary one.
Both can be extracted by using the command
tar xvvf name.tar.gz
The source tarball will extract to a directory usually with the same name as the program. You can then compile it and do a "make install". A binary will follow the directory structure that it was meant to be installed at (ex. /usr/bin/program). If it does this, copy the tar.gz file (as root) to your root directory and extract it again.