Quick and dirty answer... Basically, Cavalry makes the grand sabre charges that smash an opponent as used in the Napoleonic Wars smashing infantry, overrunning arty and routing the enemy. There just was not the room to maneuver on most battlefields in the US that there was in Europe. Cavalry doesn't function well in forests and the battlefields of the ACW were dominated by forests. Cav was basically broken up into three groups: Light, medium/standard & Heavy Cav w/ sub groups like Lancers, Hussars, Dragoons etc.
Light Cav was your recon, pursuit, screening, flanking etc. units. Frankly, almost all Cav in the US was Light Cav... because that is what it took to catch and defeat the plains tribes and it worked well. By the end of the war there were European observers that viewed the US Cav as the finest light cav in the world.
Mounted Infantry was literally just that, Infantrymen mounted on horses or mules for mobility sake, they dismounted to fight. When given rifled carbines all Cav could be used as mounted infantry and most were. This is particularly useful when using Cav to secure bridges, crossroads etc. Typically Cav could take territory but could not be expected to hold it. Largely because cav units were smaller than infantry formations. Also a typical horse will only carry a couple hundred pounds, when you subtract the man it limits how much gear is available to the individual Cavalryman.