An exoskeleton in nature is an external exoskeleton, typically seen in insects, crustaceans etc. A robotic exoskeleton is a mechanical device worn externally by a human.
An exoskeleton is generally considered to be a hard mechanical frame, with joints which allow the human operator to move. There are however soft exoskeletons which are worn over the body, and which rely on the humans internal skeleton to transfer the forces from the actuators worn over the body. A frame can be anthropomorphic (might resemble a human) or pseudo-anthropomorphic (not human like).
An exoskeleton can be powered, or passive. A powered exoskeleton can use any type of actuator including hydraulic, electric, pneumatic etc. A passive exoskeleton is unpowered.
A powered exoskeleton can be classed as assistive or enhancive, an assistive exoskeleton can help a paraplegic person who is paralysed from the waist down to walk, by walking for them. An assistive exoskeleton could also help an elderly person walk by offering say 10% of the required power to help them move. An enhancive exoskeleton could augment the strength of its wearer, allowing heavy items to be lifted easily. An enhancive exoskeleton could also help with endurance, for example walking a longer distance, or using a heavy tool for extended periods.
A passive exoskeleton can provide mechanical support to people when sitting or standing for long periods, or assist with movements using a spring to store energy. These can then be used to hold a tool above the head for extended periods.
A powered exoskeleton can be tethered or untethered – this relates to its connection to power or control. For example a hydraulic exoskeleton will generally be tethered to both a control computer and a hydraulic pump.