Truck with lifting platform
Five general types of tow trucks are in common use, generally based on the type or size of the vehicle to be towed:
Boom – Use an adjustable boom with a winch to retrieve vehicles from a ditch, over an embankment, or anywhere the vehicle cannot be safely reached by backing up. Some booms are stationary, some use heavy pivoting A-frames, others are equipped with hydraulically actuated telescopic tubes. The heavier types of boom can rotate, effectively turning the crane into a kind of mobile crane, called a “rotator,” and are generally reserved for heavy vehicle accidents.  In the past, forklifts used a “hook and chain” system in which the chains were wrapped around the frame or axle of the vehicle and then lifted using a boom winch. A tow bar with heavy rubber mats connects the truck and the vehicle, so it can be towed on its other axle. The “slings” and “belt lifters” are an evolution, with rubber straps replacing part of the chains.    Slings are not widely used today because they can scratch car bumpers. But they are sometimes used for towing vehicles that have been in an accident or have one or two of the front or rear wheels missing or for pickup trucks and other vehicles that have steel bumpers. Cars equipped with all-wheel drive cannot be towed with a sling as it can cause problems with the car’s transmission. 
Wheel lift in stowed position
Wheel lift (also called “eyeglass lift” or “bottom lift”) – Developed from hook and chain technology to produce a large metal yoke that can be placed under the front or rear wheels to cradle them, drawing the front or rear end of the vehicle clear of the ground using a pneumatic or hydraulic lift so that it can be towed. This device generally picks up the drive wheels of the vehicle (that is, the front wheels if it is front-wheel drive, the rear wheels if it is rear-wheel drive) by touching only the tires.  The wheel lift was designed by Arthur W. Nelson of Weld Built Body Co. in 1967.  The chairlift name is common in Europe; the crib resembles a pair of square spectacles (spectacles). Medium and heavy duty trucks use a variation, the “bottom lift” or “chassis lift”, which lifts the axle or frame instead of the wheels. Forklifts may have adapters that can also lift the chassis. 
Integrated (also called “Autoloader”, “Snatcher”, “Quick Pick” or “Repo Truck”): boom and wheel lift integrated into one unit. It is used in light trucks to recover vehicles or move illegally parked vehicles. Most have controls for the apparatus within the cab of the crane to allow for quick pickup without the inconvenience of exiting the truck to hitch the vehicle.   Heavy duty trucks are also built with an integrated lift.
Flat bed with wheel lift
Platform (also called “kickback”, “skid” or “tilt tray”) – The entire rear of the truck is equipped with a platform that can hydraulically tilt and slide down to ground level, allowing the vehicle to be positioned on it for its own means or pulled by a winch.  Because they transport rather than tow the vehicle, they can be used in a completely immobile vehicle; in the US they are used to transport cars heavily damaged by crashes.
Raise the flatbed
Lift platform – A boom uses a wheel lift frame to lift the vehicle vertically and load it onto the platform. Used in Europe, this truck can remove vehicles that are parked in parallel.
These are the most common arrangements, but are by no means exclusive, as there are flatbed units that offer a wheel lift, boom trucks that can retrieve but not tow, and wheel lift units that offer a combination of boom with sling.