First, it's important to remember that there are two levels of notifications you can set up — "Staff" and "Business".
Business refers to the ones set up under Settings and "Appointment Notifications".
These notifications are sent whenever anyone makes an online appointment (those made via the BookNow button/link, as opposed to those you enter into the calendar manually), with any staff member in your account. The email notification you receive will contain the full appointment details, as well as an .ics file attachment that you can download into an external calendar (Google, iCal, Outlook). An email is also sent when a client cancels an appointment with your business.
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You have the option of receiving a text (SMS) message on your cell phone, notifying you of new appointments, and containing the appointment details.
Staff notifications are set up on the Staff Details page, and sent to that staff member when he/she receives a new appointment.
If the calendar synchronization via email option is selected, the staff member receives an email with an appointment that's automatically accepted in the email address's corresponding calendar (ie: a Google Calendar, if the address is a Gmail one). It's also important to note that a notification is sent whenever any appointment is made, edited, canceled or moved — regardless of whether it was made online or offline.