Dragoons are a type of mounted infantry in Empire: Total War.
Despite appearances, dragoons are not really cavalry but mounted line infantry, riding into a battle then fighting on foot.
Horses give mobility, not shock value in combat. Dragoons do not charge home, but ride to critical spots on a battlefield where infantry firepower is needed. They carry smoothbore, muzzle-loading muskets with shortened barrels (to make reloading easier) originally called "dragons", or dragoons; over time, this name transferred to the men.
Dragoons are also useful for riot control and civil suppression ("dragooning" is to bully people into a course of action). Their usefulness as infantry and "cheap" cavalry means that they put down all kinds of trouble, as cutting down civilians is beneath the dignity of proper cavalry regiments.
Historically, dragoons slowly became cavalry soldiers like any other, and stopped fighting as mounted infantry, although many regiments did retain the name. The cavalry had always regarded them as (lower paid) social inferiors, and the infantry had resented them as not being proper footsloggers, so the dragoons welcomed their new acceptability.
Dragoons can fight with equal competence on foot and on horseback, using their carbines and sabers to cut down their enemies. As horsemen, dragoons are light cavalry, capable of defeating routing or wavering enemies, using their mobility to flank and smash their enemy. On foot, they act as a small detachment of line infantry, although they are no match for dedicated line regiments.
Dragoons are highly versatile units. The mobility of dragoons allow them to flank enemies, take ground, or to screen larger movements. On maps with commanding central high ground, they can rush forward, dismount, and hold the ground long enough for regular infantry to advance. And should the situation go against them, they can remount and escape. They can also be used to lay ambushes or flank enemy forces, dismounting and engaging with their carbines. Their mounted and dismounted stamina are calculated separately, so if they're exhausted whilst mounted, they'll be fresh when dismounted (assuming they haven't already fought when dismounted) and vice versa, their horses will also recover stamina whilst dismounted, and the men will recover stamina whilst mounted.
Dragoons benefit greatly from advanced training centers, gaining up to four chevrons of experience upon recruitment.
Unfortunately, dragoons are inferior dedicated heavy cavalry or even light cavalry in a melee; they lack the stamina of dedicated light cavalry, and they are not as capable in terms of shooting, durability or melee capabilities as most infantry. Dragoons are largely supplanted by Light Dragoons, which can fire without having to dismount while having no significant disadvantages to regular dragoons.