1) Check that the order email is receiving the confirmation: Assuming you have an email address set in the Settings > Store Settings > Company information, Order email field, that email address will also receive a confirmation. If that email address received the confirmation, this means that both emails have been sent (as the one your customer receives is simply a BCC of yours).
2) Ask the customer to check their Junk / Spam email
3) Check your email or the customers email is not on a blacklist: Use this: http://mxtoolbox.com/blacklists.aspx to check if your email is on a blacklist - Place your email address (and then the customer) in the section they ask for an IP address and a mailserver will be suggested. If you are on a blacklist, contact your whoever is managing your email (for example Ashop, or your ISP etc) and let them know about it. If your customers email is on a blacklist, contact them by phone to let them know of the issue.
4) If none of the above are working: Please contact us on live chat or email stating that you have checked the above so that we can help you troubleshoot.