Sakina,
How you do this would depend on a number of factors, some of which are:
- the web services or other API available in the course booking system
- where the integration code will live - Moodle, course reg system, separate application (i.e. some form of middleware)
Possible scenario:
- user books a course on the course booking site
- when they "finish", they are sent to the moodle site
- on login to Moodle, their moodle enrolments are checked against their bookings on the cousre booking site
- any missing enrolments are created in moodle
note, this assumes that the courses have been created and are available in Moodle before they are made bookable from the booking site
If you decide to write the integration code on the Moodle side, you should implement it as a plugin (or a set of plugins). At all costs, avoid hacking core code. Hacking core will cause you a lot of grief in future.
If you decide to write the integration code on the external site and call Moodle's web services API to register/enrol usrers into Moodle, you'll find the API documentation at http://your.moodle.url/admin/webservice/documentation.php helpful
In terms of minimum information required by Moodle to register a user, those are: username, email, firstname and lastname. In order to enrol in a course, minimum requirements are userid, the role id, and the contextid - those terms may not mean anything to you right now, but you'll become familiar with them once you start developing your integration.
Hope that helps.
Muyi