Try:
<?php
// Inserts first name and email address intot the b_request table
$connect = mysql_connect("localhost", "username", "password") or die ("Wrong password, jerk.");
mysql_select_db ("db_name") or die(mysql_error());
$query = "INSERT INTO B_request (`name`, `email`) VALUES ('".mysql_real_escape_string($_POST['name'])."', '".mysql_real_escape_string($_POST['Email'])."')";
$result = mysql_query($query) or die('Error:'.mysql_error());
?>
mysql_real_escape_string() protects you against SQL injection.
You may also want to make sure the email address is a valid one before inserting it into the database. There are plenty of threads on form validation here.
Post any error message you get.