There are different ways how a web server can handle PHP files. Following are the most common implementations how this is done:
“mod_php” is a module for the web-server “Apache”.
With this module PHP is therefore “integrated” into the web-server. So there is no extra PHP-Process which handles the PHP interpretation. Instead everything is handled by the Apache process.
The main advantage for using “mod_php” is performance. Compared to CGI you usually gain 300-500% when changing to mod_php.
Main reason for that is the ability to cache PHP modules, or specifically the configuration which usually (in CGI and FastCGI) has to be parsed on every request.
The “Common Gateway Interface” (short CGI) implementation means, that the web-server starts an extra PHP process for each request. Therefore all PHP modules, the php.ini and all other configuration needs to be loaded and parsed on every request, which is inefficient.
Main advantage for using CGI is the complete separation between the executing web-server Code and the PHP code.
FastCGI is a PHP implementation, which contains the security advantages from CGI but also being efficient like mod_php.
Here we don’t start a new PHP process on every request, instead we use “ready made” PHP interpreter instances which only get the PHP files passed on to be handled.
The „PHP-FastCGI Process Manager“ (short PHP-FPM) is an alternative to the FastCGI implementation.
Here the main difference is, that there is always a “parallel” PHP-FPM Process which is connected to the web-server process.
For more details on FPM see HERE