If your Bookings are set for manually approve bookings, there are several ways how you can find out, that a new booking was made (if this option is untagged, the new appointment will just appear in the Calendar).
In the Calendar tab a number, indicating how many new bookings have been made, will appear.
A notification about a new booking request appears above the Calendar in the Calendar tab. It includes the name of the client and the service being booked.
In the Calendar, on that specific date and time, a new slot with the clients name will appear.
You also receive an email notification about the new request.
If the client did not pick a particular staff member, you can assign him one in this window (otherwise you might be overbooked). Simply click on Not assigned in the Who? bar. Then just choose a particular staff from the list, which will automatically unfold.
At the end of the table, you will notice two buttons - blue one with Confirm and a red one with Reject. Choose the corresponding one, depending on whether you want to confirm or reject the booking.
If you reject it, a window pops-up, where you can write a polite, but clear message to your client explaining why you have rejected his booking. If you confirm it, a field with the particular booking will become fully colored, instead of being just hatched.