Installing PHP Zip Extension

You may have several php.ini files, one for CLI and one for apache. Run php --ini to see where the CLI ini location is.

This is how I installed it on my machine (ubuntu):

php 7:

sudo apt-get install php7.0-zip

php 5:

sudo apt-get install php5-zip

Edit:
Make sure to restart your server afterwards.

sudo /etc/init.d/apache2 restart or sudo service nginx restart

PS: If you are using centOS, please check above cweiske's answer
But if you are using a Debian derivated OS, this solution should help you installing php zip extension.