Trail shoes are basically designed for running off road. However, even a trail shoe will come in differing categories such as for tracks and bridleways, parks and grassland, forests and of course those which are more suited to rugged hill and fell running.
A good number of trail shoes are now also designed with enough cushioning so as to be able to mix off road with some on road too. In some cases popular road running shoes have even been converted and manufactured to allow for off road running too, but needless to say too much road running will lead to a wearing down of the extra grip which comes with these shoes.
For those who want a lighter weight trail shoe for racing, then these are also available, but many will have a lower profile similar to that of a fell shoe for extra stability. At the same time, they are less durable with of course less cushioning so think about not only the terrain which you might be racing over, but the time on your feet and distance too.