Tripods add a level of professionalism to your shots. They prevent the audience from being distracted by too much camera shake and are especially useful when zooming in to a distant subject.
Ideally your tripod will have a mini spirit level near the top. This is simply some liquid with a bubble in it, by lining up the bubble near the top of the dome it is in you will know the head of the tripod supporting the camera is level. This is useful when you are not sure whether it is you, or your subject which is on a slant!
Used well, they bring a sense of calm and stillness to a shot. A handheld camera will create tension if done well.