Although simple, URL rewrite in Joomla can be pain in neck. In order for Joomla URL rewrite to work apache should be configure to allow URL rewrite in .htaccess . If you are not sure, contact your web server administrator. You can even test yourself, see http://www.addedbytes.com/for-beginners/url-rewriting-for-beginners/ .
If you are managing the server then manually enable it. If you are not sure then follow the instruction on following link see: http://www.jarrodoberto.com/articles/2011/11/enabling-mod-rewrite-on-ubuntu
Now go to Joomla installation folder and rename htaccess.txt to .htaccess
mv htaccess.txt .htaccess
Open .htaccess and remove the comment following line (find the following and remove # in front of it):
If your Joomla is not installed in your root folder then:
Now go to global configuration in the admin panel and enable:
Search Engine Friendly URLs
Use URL rewriting
If necessary restart the apache server. Try navigating your website it should not be showing index.php.