The contact form is a PHP function which collates the data entered through the form and uses the PHP Mail function to send the email to the specified address. If you're not getting the messages, firstly check whether you can receive standard WordPress notifications, for example, try resetting your password.
If you receive WP emails, but not the contact form messages then the Mail function has not been correctly setup by your webhost. You have 2 choices, firstly to ask them to enable and configure Mail; or secondly, modify the PHP script which sends the email to use the same fucntion which send WP notifications. Open up /framework/php/class-form-generator.php and find this code (around line 239):
mail($to, $subject, $message, $header);
Change it to:
wp_mail($to, $subject, $message, $header);
If messages are being sent correctly, you should already receive a notification that the message was sent. The default is:
Your message has been sent!
You can modify the text in /includes/contact-form.php if you use another language or wish to provide more information.