armored warriors who fought on horseback during the high middle ages?

Answer 1

They were called Men-at-arms (Sometimes, Gentlemen of coat armour and other names) and includes knights, esquires and other gentlemen (&Nobles). Though they could also fight on foot their main possition was as cavalry or at least as mounted heavy infantry (as was common for the English in the 14th & 15th Centuries). Lighter armoured cavalry had other names like hobilars.
Sometimes men-at-arms are thought of as poorer foot soldiers; this is a mistake.

Answer 2

The answer you are probably looking for is "knights", although as the gentleman above points out, that is a bit simplistic when applied to the entirety of the middle ages.

However, knights were supposed to adhere to a code of "chivalry". "Chivalry" is clearly derived from the same root as "cheval" the French word for"horse". So there is a connection.

A knight could *also* in some cases be a man-at-arm -- the two terms weren't *always* mutally exclusive. Of course, by the later middle ages, knights were considered minor nobility (often Barons, Baronets or Knights Banneret) and a man-at-arms was not, even if he fought on horseback.

Hope this helps.