"Unit" can mean any 'official' organization. The standard field organization that operated as a unit is a regiment commanded by a colonel (1,000 men in 10 companies). A company is nominally 100 men commanded by a captain. Regiments could be divided into two or three battalions. Men were typically recruited and enrolled by companies and these companies usually came from the same locale -- a city or county. Depending on how many men were recruited or based on a particular need (e.g. scouting duty) they could be officially designated as a battalion. Battalions are typically commanded by a lieutenant colonel or a major. If they could get enough men to form ten companies, they could be officially designated as a regiment.

The numbers above are 'by the book'. In actual practice, units were smaller and as the war went on Confederate units could be half to a fourth of the regulation size. There were 'independent' companies which were unassigned to a battalion or regiment. These usually performed some specialized service like frontier protection or guard/garrison duty etc.. Some regiments routinely separated their battalions for various reasons such as scouting or raiding.

Military organization at the time, from largest to smallest consisted of:

Army -- General (Full General)
Corps -- Lieutenant General
Division -- Major General
Brigade -- Brigadier General
Regiment -- Colonel
Battalion -- Lieutenant Colonel or Major
Company -- Captain

Infantry are generally foot soldiers who were armed with long rifles and bayonets. Mounted infantry or "mounted rifles" are armed and supplied as infantry but, obviously, move by horseback, typically supplying their own horses.

Cavalry are, of course, mounted on horses and equipped with carbines (shorter rifles), swords, and pistols. Cavalry rarely fought on horseback. Horses were held in the rear while the men fought on foot like infantry. Cavalry in the west often brought their own horses but were paid for their horses. Dismounted cavalry is a cavalry unit who has either lost all its horses or had their horses taken away (usually because the army could not afford to feed or supply them) and/or may be waiting for replacement horses.

Again, this is 'by the book' and actual practice varied significantly. For example, west of the Mississippi, Confederate cavalry almost never had swords and might typically be carrying a sawed-off shotgun and one or more revolvers.