Put simply, bruxism is teeth grinding. More accurately, bruxism describes a range of related behaviours including teeth grinding, gnashing and clenching of the jaw. Bruxism is usually a subconscious behaviour performed whilst one is asleep.
Teeth grinding and clenching can permanently damage one's teeth as the enamel is worn away. However, bruxism leads to other problems as well (both short-term and long-term) such as headaches and disorder of the TMJ.
There have been many treatments proposed to reduce or stop teeth grinding. However few have any clinical evidence. (Those that do are occlusal splints (mouth-guards, night-guards), Mandibular Advancement Devices (MADs), botox and hypnosis.)