How to execute SQL SELECT query and see the results?

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.