When the customer pays, open the invoice and press Record payment (amount, date, method, reference), or Mark as paid to record everything still due in one payment. To remind them, press Email again. The Invoices page header shows the total waiting to be paid.
If an invoice was wrong, press Void: its jobs return to Ready to invoice so you can invoice them again. Delete draft deletes a draft and puts its jobs back. An invoice with a credit note in Refund due cannot be voided until the refund is marked done or the credit note is voided. To take off only part of an invoice, use Add credit note instead.