This form of military headgear gained popularity in the 18th century, as a badge of honor for elite units. After all, the bear is a fierce fighter. Grenadier company's in the 18th century were often equipped with bearskins. Historically, a grenadier was a soldier who carried grenades into battle. Not grenades in the modern sense, but rather hand held bombs. The soldier lit the bomb's fuse with a slow match (slow burning, twisted cord) carried in a metallic vessel attached to his cross belt and threw in into the enemy's ranks.
By the 18th century, grenadiers no longer carried hand bombs. However, they were an elite flank company in a regiment. Grenadiers were tall and imposing. As heavy cavalry served to role of "shock troops," so did grenadiers. Already tall men, the bearskins increased their height to make then more imposing. The British monarchy ordered its grenadiers to wear the bearskin in 1768. The Don Troiani painting depicted below is a good representation of a British 18th century grenadier company (this one from the 23rd Royal Welsh Fusiliers) in service during the American Revolution.
By the Napoleonic Wars, grenadier companies in regiments ceased to exist. The British military seems to have fine away with the designation altogether. In the the army of Napoleon I, however, the title remained on a regimental basis, but with limited wearing of the Bearskin. The most notable of these regiments being the Imperial Grenadier Guard regiments. Fondly referred to his "Old Guard," the men comprising these regiments were older and experienced -- having fought in the early battles of the wars. They were typically of larger stature and wore an excessively tall bearskin bonnet. Napoleon sent in the Old Guard to apply the coup de gras to his opponents.
For the most part, the British army during the Napoleonic Wars did not wear the bearskin. One cavalry regiment, however, is an exception. The 2nd Royal North British Dragoons, or Scots Greys, wore a bearskin. Queen Anne bestowed the honour of wearing the bearskin upon the regiment in 1706 due to its service during the Battle of Ramillies. The regiment is most famous for its charge of the French Grand Battery at Waterloo, as depicted by the portion Lady Jane Grey's massive painting of the event depicted below.
After the Battle of Waterloo, George III awarded the 1st Regiment of Royal Foot Guards the title "Grenadier" and the honor of wearing the bearskin. The King bestowed this honour to reward the 1st Foot Guards for their role in defeating Napoleon' Grenadier Guards. In the 1830s, the monarchy extended the honor of the bearskin to the other regiments of the Household Guards Division: the Coldstream Guards (never call a member of this regiment a guardsman, they are referred to as Coldstreamers); the Scots Guards; the Irish Guards; and the Welsh Guards. The regiments' bearskins are distinguished by whether or not they are adorned with a hackle (a horsehair plume), the color of the hackle and the side of the bearskin upon which the hackle is worn. Of course, officers' bearskins are taller and more ornamented that those of the other ranks.
Traditionally, British bearskins are made of the fur of the male black Canadian bear. However, the fur of the female brown Canadian bear, dyed black, has also been used. The fur is mounted on a wicker frame with a leather hat band. There is no visor.
The use of real fur has been the subject of debate over the years with the disagreement not limited to modern day political correctness. In the late 1800s, there was an outcry from Americans opposing the killing of bears to make bearskins due to the dwindling animal population. Looking online, most recently a push to abolish real bearskin use occurred in 2008. As of 2012, the British Ministry of Defense has bee. considering a faux alternative.
18th or 19th bearskins are not readily found in the militaria market. There are reproductions, but they come with a fairly high price tag. I was fortunate enough to own a 1930s Coldstreamers bearskin a few years ago. As a sidenote, it is said that the hair on the skin continues to grow. I don't know if that is true or not, but the image below, posted by a friend of mine on Facebook, tends to support the assertion.