A variety of snares, deadfalls, and drag nooses can be constructed from common materials, even simple vegetation found in the forest or woods. Usually, wire or some kind of rope or string will help a trapper to build a better trap, but you have to do what you have to do!
A simple snare can be constructed by placing a noose over the trail or den opening of an animal. You must then attach the noose to a secured location so that the pray cannot simply drag the rope or string along with it. This trap will tighten around the animals neck as it pulls through, and will trap the animal by not allowing it to leave, sort of like a dog on a leash. This is not a lethal trap, and sometimes the animal will be able to get away. Wire is a good idea on a simple snare as opposed to string, and is most certainly better than vine or something like that.
When setting a trap, it is important to mask your human scent so that animals will not be weary of the trap you have set. Nothing will scare an animal off faster than the smell of a human. There are several ways in which you can go about doing this. For one, you can hold the trap over smoke before you go to set it up. Smoke will mask a human scent very well. You could also cover your hands in dirt, especially dirt that smells like rotting vegetation. This will mask the scent on your hands, allowing you to set the trap without making it smell like you. Also, try not to disturb the area too much, as animals know their habitat quite well, and are more aware of what is gong on than people think!