Trail running has developed into a pretty loose term, ranging in exact meaning from runner to runner. Concretely it is running off-road. It has some benefits over road running, but also has its own set of risks as well.
Off-road running, depending on the type of terrain, can be a bit easier on the body, especially the knees and feet. This is more the case for light or medium trails with little to no half buried rocks or roots. The softer ground is definitely more forgiving and offers less shock from impact.
Although it has the pluses mentioned above, the risks are different than what you might encounter on the road. When you take it off-road, there is a chance to trip over a root or other debris in your path, which can cause some significant injuries. Uneven ground is also something that you need to pay attention to. If you're used to running on an even surface like a road, you'll have to pay attention to your feet until you're more accustomed to the different terrain.
The difference for most is the chance to put running together with nature. You won't need to worry about traffic or being hit by a car. You can simply take in your surroundings while you go, often translating to a longer run, as your mind isn't constantly on what your feet and legs are doing.