Suspension hanging where the executee is lifted into the air using a crane or other mechanism. Death is caused in the same way as with short drop hanging.

Standard drop hanging where the prisoner drops a predetermined amount, typically 4-6 feet, which may or may not break their neck. This was the normal method adopted in America in the later 19th and early 20th centuries.

Finally, measured or long drop hanging which became universal in Britain from 1874, where the distance the person falls when the trapdoors open is calculated according to their weight, height and physique and is designed to break the neck. This method was adopted in British Colonies and by some other countries who wished to make executions more humane.
Each of these processes are examined in detail below.

The "Short Drop" method.
Hanging using little or no drop is still used by some Middle Eastern countries, notably, Iran. It is far more common in the 21st century than long drop hanging.
Short drop hanging was effectively universal up to around 1850 and was usually carried out in public. The prisoner could be suspended by a variety of means, from the back of a cart (or later a motor vehicle), from a horse as was sometimes used in America, or by removing the platform on which they stood, as was used in Nazi hangings and also in present day Iranian ones carried out inside prisons, or by some form of trap door drop mechanism as was used in Britain from 1760 and adopted by many other countries.
This 1809 picture of the triple hanging on the New Drop gallows outside the Debtor's Door of Newgate in London shows clearly how little drop was given at that time.

Suspension hanging.
This method is currently used in Iran for public hangings and was also used for some executions when the Taliban controlled Afghanistan where executees were hanged from the barrels of tanks and from mobile crane jibs. In Iran, both mobile crane and recovery truck jibs are used. All of these have hydraulic mechanisms for raising them, so the jib serves as both the gallows and the means of getting the prisoner suspended.
In America, instead of the conventional gallows that dropped the prisoner through a trapdoor, some states used a method where weights connected to the rope jerked the person upwards when they were released by the hangman. This method was used in 1874, for the hanging of William E. Udderzook in West Chester, Pennsylvania and also for Charles Thiede in Utah in 1896.
Connecticut used a similar arrangement for the execution of Gerald Chapman on April the 26th, 1926. A weight was connected to the rope which passed over a pulley. The warden operated a lever with his foot to allow the weight to fall, so pulling Chapman 12 feet into the air with such force that his neck was broken.In New York state Roxalana Druze was rather less fortunate when she was hanged for the murder of her husband in 1887.As a result New York developed electrocution as its method of execution although men continued to be hanged there until the end of 1889.

Standard drop hanging.
A standardised drop, of between four and six feet, was used in many American hangings during the later part of the 19th century and into the early 20th century. This was not worked out against the weight of the individual, but was often equivalent to their height. It was considered as an advance on the short drop method previously used. A drop of this distance was often not sufficient to break the prisoner's neck, however, and many still died by strangulation, although in a lot of cases they were knocked unconscious by the force of the drop and the impact of the heavy coiled knot against the side of the neck. Occasionally, they were decapitated when the drop proved to be too long, as happened at the execution of Eva Dugan in Arizona in 1928. Standard drops were given to the eleven senior Nazis executed after the Nuremberg trials and several were reported to have died slowly. The Lincoln conspirators were given a drop of five feet at their hanging in 1865 and at least two of the four struggled for some time after they were suspended. A number of hangings of German war criminals carried out under American jurisdiction at Landsberg and Bruchsal in Germany were filmed.Some of these films show the hangman attaching the rope to the beam so that the noose was laying on the trap doors.This resulted a in drop roughly equivalent to the height of the prisoner.In the film clips there is seldom obvious signs of struggling, e.g. wild vibrations of the rope.