The most common cause for emails not being delivered is usually a server configuration problem. To rule out any server issues, install the free Check Email plugin available on wordpress.org and try sending yourself an email. If you don’t receive the email, contact your hosting provider, they should be able to help.
Sometimes emails get caught in spam filters. To improve email deliverability try setting the From email address to an email from the same domain as your website (Eg. if your website is holidayhome.com, set the email firstname.lastname@example.org). If you want customers to reply to a different email address, you can set your personal email in the Reply To field. Including the customer’s name in the email also helps. You can add this using email tags.