Clean URLs are very useful for SEO and make a site look a bit more elegant, as well as giving the users a sense of where they are even if its not a physical location.
The trick to getting clean URLs to work on 1and1 is to go into Drupal's .htaccess file and uncomment the following line:
# RewriteBase /
should be
RewriteBase /
or if you have an older version of Drupal:
# RewriteBase /drupal
should be
RewriteBase /
You have to do this because your site is running in a VirtualDocumentRoot at http://example.com