This question arose in my mind today that when you already have the fuse for safety precautions why do you need the earthing?
Fuses are typically to protect wiring and other components from overheating and perhaps catching fire and/or exploding in the event of a low-impedance fault. From the Wikipedia article "Fuse (electrical)":
A fuse interrupts an excessive current so that further damage by overheating or fire is prevented. Wiring regulations often define a maximum fuse current rating for particular circuits. Overcurrent protection devices are essential in electrical systems to limit threats to human life and property damage.
However, one could easily be electrocuted without tripping a breaker or blowing a fuse. Thus, earthing a metal appliance, enclosure, plumbing, etc. prevents potentially lethal voltages from developing on these conductors.
If the live becomes in contact with any exposed metal part of a device and you touch that metal part it would not be good news.
If the exposed metal parts are connected to the Earth then a very low resistance circuit is completed when the live touches the metal part.
A very large current will flow, blow the fuse and disconnect the live from the device.
Some devices with metal parts do not require an earth because all the metal parts are covered with an insulator. This is called "double insulation".
They do different things. Almost opposite, in fact.
Earthing prevents potential difference building up between the device and surroundings (which would flow through you if you touched it, or arc through the air, or to the nearest chip or sensitive component, blowing the device).
The fuse prevents a large current flowing through the circuit. In a lot of cases, there is a breach from the live wire to the casing, and it's the earth wire that ultimately swallows the current.
So, you need the earth to equalize potentials, which prevents the current that would blow the fuse. And you need a fuse to prevent a large current flowing through the earth (and other parts too).
Fuses are over-current protection - a 10A fuse might help prevent a fire, since it limits the power that an appliance can receive.
What other answers haven't mentioned is earthing or protective-earth wiring (PE). Sometimes a live wire may come in contact with a metal casing of a device - this will not cause a fuse to blow. However the case is "live" and if someone were to touch it the results might be fatal. That's where a residual current device comes into play (https://en.wikipedia.org/wiki/Residual-current_device). Current should normally run only in live or neutral wires. If it happens to run in a PE wire, or finds another path to ground (e.g. through a human body) the RCD will break the circuit. This may happen at current of 10-30mA flowing the "wrong way" even though the appliance nay draw 10A and function normally.
Personally I consider the RCD to be a simple and beautiful device - go read more about it ;)
To add to the previous answers, a sustained current of around 30mA is enough to kill a person. The fuses in house circuits might be in a range around 5A to 45A. So a fuse by itself does nothing to prevent electrocution.
The earth (ground) conductor provides a low-resistance return path for current should a live wire come adrift inside an appliance and touch a metal part. This low resistance will allow a very high current to flow, blowing the fuse. This cuts off the power before anyone touching the appliance gets electrocuted.
Though the FUSE breaks, there is a chance of "arcing" which then needs to be safely "grounded".
FUSE: A electrical conductor with very low resistance. EARTHING: A safe termination for the OverCurrent which may appear in a circuit due to various anomalies.
The Low Resistance offers protection by getting itself destroyed in the process of safeguarding the equipment in line. When overcurrent scenario occurs, the Low Resistance melts, hence breaks the circuit.
Before melting and breaking the circuit, any amount of spiked up current is safely grounded using the Earthing terminal. This is not the end of it.
If the over current is so huge, and the gap created by fuse may be too little. This may result in an "Arc" which is similar to lightning during storm in visibility.