Search Engine Friendly (SEF) URLs changes the way site links are presented and optimises them so that search engines can access more of your site. Be aware of that this option is only available to sites hosted on Apache servers.

Five steps to turn on SEF support

1. You must run Apache powered webserver
2. You must have mod_rewrite enabled
3. Your server must allow .htaccess override
4. Search Engine Friendly URLs must set to 'YES' ($mosConfig_sef="1" in configuration.php)
5. Rename htaccess.txt to .htaccess
   
   

Troubleshooting in case of a failure

1. Joomla Administration -> System -> Sytem Info -> Tab: System Info:
   Webserver type is listeted next to "Web Server"
2. Joomla Administration -> System -> Sytem Info -> Tab: PHP Info:
   In section "Loaded Modules" mod_rewrite should be listed here.
   To enable mod_rewrite with apache2 (Debian):
   
   • cp /etc/apache2/mods-available/rewrite.load /etc/apache2/mods-enable
   • /etc/init.d/apache2 restart
3. Saka from the Joomla Team suggested in his FAQ for testing the server override:
   "To check this delete everything from your .htaccess file and write a simple command here, for example:
   
   'Redirect /google.html http://www.google.com'
   Now point your browser to http://www.yoursite.com/google.html
   If it redirects you to Google.com then you are OK."
   
   To enable .htaccess override with apache2 (Debian) add followin lines to your apache2.conf:
   <Directory /path/to/joomla>
       AllowOverride All
   </Directory>
4. Joomla Administration -> System -> Sytem Info -> Tab: System Info:
   Section "Configuration File" ->  $mosConfig_sef="1"
5. To rename the file
   
   • mv htaccess.txt .htaccess
   or with a ftp program of your choice...

Finally the result 

before:
http://www.hoernchenpower.org/index.php?option=com_content&task=category&sectionid=3&id=17&Itemid=34

afterwards:
http://www.hoernchenpower.org/content/category/3/17/34/