Mongkol (pronounced "Mong- kon"):
First off, the Mongkol is unique to Muay Thai, and is not used in Cambodia or Burma (which have very similar arts, such as Pradal Serey); that particular fact is from Wikipedia lol. The Mongkol is a often described as a "crown". It is a sacred (blessed by a monk) piece used to represent the gym or camp the fighter represents.
In the past, each camp had their own distinct Mongkols, but now gyms often have more than one. Fighters do not touch it; It is placed (before the fight) and removed (after the Ram Muay) by their Kru Muay.
These arm bands worn around the bicep are meant to protect the fighter from harm. They are worn throughout the whole fight. They are given to the fighter by friend, family etc