gzip is short for GNU Zip, a GNU open-source replacement for the Unix 'compress' program.
The usual file extension for gzipped files is .gz. Unix software is often distributed as files ending with .tar.gz, called tarballs. They are files first packaged with tar and then compressed with gzip. They can be decompressed with gzip -d file.tar.gz. Nowadays more and more software is instead distributed as .tar.bz2 archives because bzip2 compresses files better than gzip.