Today, probably the most used and with less overhead is SAPI: http://en.wikipedia.org/wiki/Server_Application_Programming_Interface
Some examples of SAPI is ISAPI that is mostly for IIS, Apache modules for Apache, and the list extends with Servers.
ISAPI: Uses a system dynamic linked library (DLL) so it "attach" to the webserver and functions can be called direct on it. http://en.wikipedia.org/wiki/Internet_Server_Application_Programming_Interface
CGI: http://www.w3.org/CGI/ normally a new process is created for each request, the data from the server is normally passed in stdin and the program writes to stdout, some information is passed as environment variables.
FastCGI: http://www.fastcgi.com/drupal/ As the name suggest, its like CGI but does not need to run the program every time a request is made, being faster and using less resources.
Xitami have their own too, called LRWP http://legacy.imatix.com/html/xitami/index12.htm
But each server can implement their own.
Note: The module or external program is what parses the script. The SAPI, CGI, FastCGI and whatever will integrate with the interpreter of the script, a binary. The interpreter then receive the request path, find the script file, and parse it. Ex: PHP has its apache and IIS modules.