A London chauffeur service is pre-booked ground transport at a price agreed in advance, provided by a private hire operator licensed by Transport for London. Private hire vehicles may only be booked through a licensed operator. They cannot legally be hailed in the street or picked up from a rank, which is what separates them from a black taxi.
That distinction matters more here than in most cities, because London layers road charges on top of one another. A single journey can attract the Congestion Charge, the ULEZ charge, a tunnel toll and an airport drop-off fee, each administered separately and each with its own payment deadline and penalty. On a pre-booked chauffeur booking, all of them sit inside the number you were quoted, and none of them reach you.