$query_birthdate = "SELECT day(user_birthday) as Day FROM users WHERE username='kkjay'";
$birthdate = mysql_query($query_birthdate, $connections) or die(mysql_error());
$row_birthdate = mysql_fetch_assoc($birthdate)
echo $row_birthdate['Day'];
Always use the mysql functions in your queries when you can, as you have done, the database server has that functionality for a reason. No reason to reinvent the wheel.
This only works because you're returning 1 row in the query. If you were returning multiple rows, you would have to loop over the $result to get each row's value.
Finally, if you're working with mysql 4.1 or greater you should NOT be using the mysql extension and really should be using the mysqli or PDO extensions to access the database. These allow access to the newer features provided by the mysql database server.