Starting from Magento 1.9 the order confirmation emails are not sent during checkout anymore, instead they are sent with the Cron. To verify this works properly:
make sure your system Cron is up and running (you can check the Cron logs: /var/log/cron
to verify that).
make sure Magento Cron is setup correctly and is running every some minutes. You should see something like this in the Crontab of your system:
*/5 * * * * /public_html/cron.sh
This schedules a task to run cron.sh
every five minute. (More reading)
There are good extensions like AOE Scheduler in Magento that help you to monitor and manage Cron jobs.
An alternative way to this is to disable the Cron for these kind of emails (order confirmation). To do this you can go to this path:
public_html/app/code/core/Mage/Sales/Model/Order.php
Copy that file and bring it to this path (if the path doesn't exist create it):
public_html/app/code/local/Mage/Sales/Model/Order.php
And then change this line:
$mailer->setQueue($emailQueue)->send();
To:
$mailer->send();
However I recommend spending some time to setup the Cron instead. I think if they wanted to use Cron for these emails it's for a reason.