Crane is a vague term, often applied loosely to any lifting device. Even those with the typical “T” shape, familiar from construction sites, come in a wide range of sizes and can be powered electrically, hydraulically, manually or even by steam. Lifting devices of one kind or another can also be mounted on helicopters, ships, trains, and caterpillar tracks.
A common feature of cranes, hoists and davits (the articulated kind seen on ships to lower lifeboats) is their dependence on cables and pulleys. Most also have gears, locks, brakes, counterweights, devices to articulate the hook, and so on.
In practice, what most matters to the client is how much it can lift, how far it can move it, and how it is going to access the location where it is needed.
Most tower cranes are anchored to the ground, although they can also be mounted on vehicles. The main lifting arm extends in one direction from a central tower while an opposing one carries a counterweight (usually concrete). The operator usually sits in a central cabin for maximum visibility, but remote control is also possible.
A davit crane is a double or single mechanical arm with a winch, originally used for lowering and raising things onto a vessel (including lifeboats). Their goose-neck shape helps them glide or skate objects above obstructions. Although still most common on ships, docks and in offshore wind turbine construction, they also have many applications on land, (read more at https://www.wisegeek.com/what-is-a-davit.htm).
Their name derives from their caterpillar tracks. Crawlers are obviously useful on uneven sites or soft boggy ground. Their broad tracks also help to make them very stable when lifting. Although some are very large, they can also be small enough to squeeze through a garden gate. They are sometimes used in graveyards because they do not damage the site.
These are made of concentric tubular booms that can extend to alter their height and reach. They are very useful in confined spaces and are often used by rescue services.
These consist of three moving beams, articulated so that the hook position always remains at the same height. They are commonly used in ship-building.
The lifting gear is supported by an overhead gantry or rails. This is the kind usually seen in shipyards moving containers.