The difference between high voltage and low voltage depends on the situation and on the field of science or industry involved. Laypeople generally consider mains to be high voltage largely because it is the highest voltage they normally encounter. The UK's Institution of Electrical Engineers defines high voltage as more than 1kV, low voltage as above 50V but below 1kV and extra low voltage (ELV) as below 50V. These definitions will be used in the rest of this article except where otherwise stated.
Whilst mains voltages are capable of delivering small sparks and fatal shocks, they cannot jump significant distances, so they are dangerous only if touched. Even then, a danger only exists if the victim is grounded; professionals will occasionally allow themselves to "float with" a device, but this absolutely requires taking special precautions outside the scope of the layperson.
Safety and insurance industry
Various safety and insurance organizations consider anything outside of the ELV range to be dangerous and in need of regulation. Voltages above this range are sometimes capable of producing heart fibrillation if they produce electric currents in body tissues which happen to pass through the chest area. The electrocution danger is mostly determined by the low conductivity of dry human skin, and if skin is wet (especially with electrolytes, including sea water) or if there are wounds, or if the voltage is applied to electrodes which penetrate through the skin, then even voltages far below 40V can be lethally high. On the other hand, voltages above approximately 500V have a natural defibrillating effect, so sometimes a higher voltage can be safer than a lower voltage. A DC circuit may be especially dangerous because it will cause muscles to lock around the wire.
Sparks in air
High voltages, i.e. strong electric fields produce violet-colored corona discharge in air, as well as visible sparks. Voltages below about 500-700 volts cannot produce easily visible sparks or glows, so by this rule these voltages are 'low.' However, under conditions of low atmospheric pressure, or in an environment of noble gas such as argon, neon, etc., sparks appear at much lower voltages. 500 to 700 volts is not a fixed minimum for producing sparks, but it is a rule of thumb. Note that small sparks caused by exploding metal surfaces can occur with much smaller voltages; try tapping the terminals of a 9V battery with a paper clip in a dark room.
Science classroom devices
A high voltage is not necessarily dangerous. Physics demonstration devices such as Van de Graaff generators and Wimshurst machines can produce voltages approaching one million volts, yet at worst they deliver a brief sting. During the discharge, these machines apply high voltage to the body for only a millionth of a second or less. The discharge may involve extremely high power over very short periods, but in order to produce heart fibrillation, an electric power supply must produce a significant current in the heart muscle continuing for many milliseconds, and must deposit a total energy in the range of at least millijoules or higher. Alternatively, it must deliver enough energy to damage tissue through heating. Since the duration of the discharge is brief, it generates far less heat (spread over time) than a mobile phone.
Note that Tesla coils are a special case, and touching them is not recommended. Among other issues, they have a tendency to arch to their own bottom-end circuitry, which can introduce 60 Hz currents at lethally high voltages to the body.
The terminals of DC high voltage machines can attract dust, lint, and bits of paper. On an everyday scale, voltages higher than a few thousand volts are required in order to create an electric field with a gradient large enough to produce obvious forces. On the other hand, the forces depend on the distance from the electrodes and the electrode shapes, and at the microscopic scale of MEMS, even a few tens of volts acts like a high voltage.
In power grids, the optimum voltage for long distance distribution usually falls well into the 'high voltage' range. Electric energy flow (i.e. power) is the product of voltage and current. Higher voltage at lower current can give the same energy flow as lower voltage at higher current. However, because of Ohm's law, the heating of the wires and the waste of energy is proportional only to current. Utility companies avoid wasting energy by transmitting it at low currents, but at high (sometimes extremely high) voltages. It is easy to trade voltage for current using a transformer, but transformers require that alternating current is used. This was the reason why Edison's DC systems were abandoned in favour of Tesla's AC systems. Today even DC voltages can be stepped up and down fairly easily by using a switched-mode power supply. However these are a fairly recent introduction and still have some problems at extremely high powers and tend to be less efficient than transformers, so AC has remained the rule for power distribution. DC is however used as the main power source for most electronic circuits (including consumer electronics equipment) and low power devices controlled by those circuits. The incoming AC power to such systems is sometimes voltage converted before being converted to DC. In other systems it is rectified to DC first and then converted with switched mode circuitry.
HVDC is now used to transmit electricity over long distances, especially where underground or undersea cables with a relatively high capacitance are in use. It is also used to transmit power between unsynchronised power grids. Separately controlled grids help prevent cascading blackouts as well as having political advantages (countries generally prefer to control their own infrastructure).