I don't know what "user registration" might entail, but if it's simply a form, you'd access the form elements in the $_POST superglobal array (manual reference: [man]variables.predefined[/man]), [man]fopen/man a file (probably with a 'w' mode), [man]fwrite/man the data you desire, and [man]fclose[/man] the file pointer.
It's as easy as that!