Earthing & Bonding
Earthing ensures that in the event of a fault, adequate fault current will flow causing rapid operation of a Circuit Protective Device (fuse, circuit breaker, or RCD) promptly disconnecting the supply. This limits the duration of any shock that one might receive, dramatically reducing the risk of serious injury or death.
During fault conditions, earthing may also reduce the voltage rise of anything earthed, which in addition to the limiting of the shock duration described above can also reduce the shock risk.
Main bonding is the electrical interconnection of incoming (metallic) services (e.g. water, gas, and oil pipes) plus any extraneous conductive parts of a building (like the metal framework used in some buildings) to the main electrical earth. This ensures that under fault conditions things like pipework running through a building are not able to take on a dramatically different electrical potential to that of the installation's earth connection. This also covers the fault situation where it is the installations electrical earth itself that is introducing the dangerous voltage.
Note that if the main bonding is not present, it is considered to be a serious electrical fault. As a result anyone carrying out modifications to any part of an electrical installation should also check and rectify any faults in this area at the same time.
Supplementary, or cross bonding is usually found in special locations containing a bath or shower. Unlike earthing it is not designed to clear a fault. What it does is electrically tie together all accessible conductive parts (pipes, taps, electrical appliances etc) that could under fault conditions introduce a dangerous potential (voltage) into the room.