A mouthguard is a specially made, rubber-like cover that fits exactly over your teeth and gums, cushioning them and protecting them from damage.
A bite guard is also known as stress guard, teeth guard, dental guard or night guard. It is a dental appliance that can be provided by the dentist to protect your teeth from excessive grinding or clenching. As the grinding/clenching generally occurs at night mostly during sleep, the bite guard is often worn during the night. Your dentist takes an impression of the top and bottom teeth from which a dental technician can construct the bite guard. The bite guard is then made from acrylic that is shaped to fit your teeth. The dentist then makes sure the bite guard is tight enough on your teeth without being uncomfortable.
A gum shield is also known as a sports guard or a mouthguard. It is an appliance you wear if you run the risk of getting hit on the mouth while engaging in sports such as boxing, football, rugby, martial arts, hockey, etc. It is designed to protect the teeth and the surrounding tissue from trauma. The rubbery plastic material absorbs the pressure and distributes it evenly over the whole jaw, minimising the risk of injury.