An 'off-book' trade is one in which shares are traded outside of an exchange or regulated body. They are typically carried out through the over-the-counter (OTC) market. Off-book transactions are made between two parties directly, outside of the order books.
After the two parties have agreed on a price, it is customary for one of the participants to report the trade and its parameters to the exchange - and wider market - in order to bring the execution 'on exchange.'
When you trade on an exchange, the transaction is recorded directly on the order book, increasing visibility for other market participants.