Open Customers and press the bin on the customer's row, or Delete on their profile in the FullDiary app. The owner can, and so can any login you have allowed to delete customers, which a manager can by default.
What goes: their private notes, the forms they filled in, their visit records and your conversation with them. What stays: their bookings stay in the diary, and any invoices and quotes you have already made keep the details printed on them.
If you only want to stop emailing someone offers, you do not need to delete them: switch offers off for that person from their booking in the diary, and they still get confirmations and reminders. Anyone can also stop your offers with one tap on the link at the foot of an email.