cavalryman, but he/she might have another name that his/her parents and or his/her friends call him/her. hussar, dragoon, cuirassier, carabinier, uhlan are all types of cavalryman. There are more...
A cavalry soldier is called a Trooper.
cavalryman
A hostler or ostler is a groom or stableman, who is employed in a stable to take care of horses. They might be treated for injuries or illnesses by a veterinarian, and the condition of their feet and shoes was handled by a farrier. However, day to day care of the cavalry mount was by the soldier that rode the horse. A knight would be aided by his squire.
A Mercenary
the Norman army had three types of soldier; spear men cavalry (horsemen) and archers
The Latin word for military is militaris. The Latin word for army is exercitus and soldier is miles. The legion was an army corps.
legatus
They are usually called the cavalry, and it is when a soldier rides to battle on horseback, and usually the soldier will fight on horseback.
soldier
Specials ops serving in Afghanistan are doing that now. You're probably thinking of a knight, although your description is inaccurate - Knights were not soldiers; they were members of the nobility.
Cavalry
In the US, prior to 1942, the US Cavalry was a mounted arm (horse units). It's missions were to exploit gaps made in enemy lines, conduct screening/security, and reconnaissance missions. The last US Horse Cavalry unit to fight from horse back was the US 26th Cavalry, which attacked Japanese light tanks in the Philippines during WWII in 1942. Congress officially de-activated (retired) the US Cavalry (Horse) in 1926. Post 1942, armor (tanks) have replaced the horse cavalry.
Cavalry.
cavalry
An auxiliary soldier was a non-citizen soldier in the Roman army who served mainly in support roles such as engineering, supply, and cavalry. They were recruited from provinces and allied countries to supplement the Roman legions.
African-American U.S. Cavalry soldiers
buffalo soldier
Cavalry.
A sentry is a name for a soldier on guard.