[source: coyotetrappingschool.com]
Warning: This post does contain discussion of a device used to inflict varying levels of pain on animals.
I have basically no experience actually using these traps, but I know some things due to the frequency with which they are used in whump fiction. Here's what I've got for you:
Bear traps are a type of trap called a "foothold trap". The intention of a foothold trap is that when an animal steps on a loaded plate within the trap, the jaws of the trap close tightly on the animal's leg, preventing it from being able to escape. (Fun fact: they were originally created to trap human poachers/trespassers in the 1600's. It was only later that we started using them for animals.)
There are many different sizes of foothold trap depending on the animal the trapper is trying to catch. This includes size 1 traps intended to catch small animals like weasels (the whole trap can fit in a person's hand), all the way to size 16 designed for very large bears. The smallest ones that might harm a human are size 3-4, generally used for bobcats or coyotes.
There are also several types of jaws for foothold traps. The older, traditional ones have triangular metal "teeth" that touch point-to-point when the jaws of the trap are closed. This type is likely to do the most damage- a large trap could easily cause a fracture or partial amputation in a human arm or leg assuming the trap was large enough to create that level of force.
More common today are offset traps, where the teeth, if present, are offset. If teeth are not present (common in smaller traps), the jaws don't quite touch, leaving a little extra space. These are intended to do slightly less damage, however that may be relative depending on the size of the trap vs the size of the human.
There are also laminated traps, where the jaws are wider to spread out the weight and decrease damage to the animal's limb, as well as padded traps, where a rubber liner on the jaws further decreases damage.
You'll have to decide if you want a size 4 trap with padded jaws (maybe a line of bruise on a small human ankle or wrist), or an antique size 16 steel bear trap that could cause a traumatic amputation, or something in the middle. The sky's the limit, as they say.
Despite the damage they can do, they are reasonably easy to reset. The jaws release when the leavers are pushed downward so they become flat to the ground. Here is a demonstration of how to open a trap by pushing down leavers:
This would be something a person could do with their hands for a smaller trap, while a larger one might require a second person to stand on the leavers to successfully release the trap.
As for moving traps, this also may work: traps are usually chained to a immovable object like a large tree to prevent the animal from hopping away. Humans could in theory untie the chain (unless, say, it had a lock on it or something).
First aid for this obviously depends on the damage done by the trap. For more on this, I recommend this post.
