It depends on the archive. Sometimes the archive contains pre-compiled binaries, in which case you'd just need to copy the extracted directory to some common location (say, /usr/local/), and then symlink the executable into your path (say, /usr/local/bin/). Sometimes the archive just contains the source, and you have to compile and install it. In this case, there is probably a README or INSTALL file that will give you instructions on what to do.
In either case, you can extract it with:
mv file.tar.gz temp
tar -zxvf file.tar.gz
Replacing "file.tar.gz" with the actual filename.
Once it's extracted, cd into the directory and see what's there.