I have searched but don't know what I am looking for.

My site is monpage.com. I would like to use tumblr as a blog service for my site. The URL would be monpage.tumblr.com. At tumblr, there offers a domain mapping service. So:

Yesterday, I set up a http://blog.monpage.com to point to a newly setup http://monpage.tumblr.com. Following instructions, all I did was that I have to have the A-record for blog.monpage.com pointing to their IP.

My question is how do they do that? I would like to provide the same service to my clients. Thanks!

It seems that http://monpage.tumblr.com is being redirected to http://blog.monpage.com/. This can be done by putting in the web directory a PHP, Perl, etc., file that performs the redirection. With PHP it whould be something like:
<?
header("Location: http://blog.monpage.com/");
?>

At DNS level this can be done by setting a CNAME record:

moonpage IN CNAME blog.monpage.com.