You're looking to create a custom A record.
I'm pretty sure that you can use wildcards when specifying A records which would let you do something like this:
*.mywebsite.com IN A 127.0.0.1
127.0.0.1 would be the IP address of your webserver. The method of actually adding the record will depend on your host.
Doing it like http://mywebsite.com/user would be a lot easier to set up if it's an option.
Then you could just add a .htaccess file that looks like this:
Options +FollowSymLinks
RewriteEngine On
RewriteRule ^([aA-zZ])$ dostuff.php?username=$1
In the above, usernames are limited to the characters a-z
The rewrite rule for grabbing the subdomain would look like this:
RewriteCond %{HTTP_HOST} ^(^.*)\.mywebsite.com
RewriteRule (.*) dostuff.php?username=%1