To get a notification whenever someone registers for a class, you will configure a BCC to be delivered to the account(s) of your choice.
Under Configure | Preferences | Payment and Transaction Configuration, click Edit and add an email address to the BCC field. Any time someone registers online, an email confirmation will be sent to this address. The setting accommodates 50 characters, so you can typically enter two email addresses. Remember that this address will receive a copy of every transaction, so it might be a good idea to set up a dedicated email account to receive these confirmation emails.
NOTE! Some schools/businesses/organizations have very strict anti-SPAM and firewall restrictions. Sometimes these restrictions will change/become more stringent. If you stop receiving confirmation BCCs, check with your IT department to make sure that there isn't something blocking receipt of these BCC emails.