I want to run a SELECT query on the Wordpress database and get the results on the screen. Any widget which does it?
Have a look at the $wpdb class, it does exactly what you want:
$myrows = $wpdb->get_results( "SELECT id, name FROM mytable" );
Try to avoid SQL queries for basic operations though. WordPress offers you many convenience functions for 80% of the tasks you would every come up with. They provide a nicer interface and preserve upwards compatibility. For example to get posts from the DB you could use the WP_Query class or the get_posts() function.