Edit: Also Chevelegers (I know I spelled that wrong.)
Cavalry of the mid-18th to the late 19th century can be divided into four rough, surprisingly blurry categories.
Light cavalry:
This includes hussars (a type of flashily-dressed cavalry originating from Hungary) and chassuers a cheval (a type of French light cavalry). Most light cavalry units carried the same basic weapons: a light curved saber for slashes and some firearms (typically a carbine and/or a pair of long-barrelled holster pistols they carried in their saddle). In general, light cavalry units were made of smaller men on smaller, lighter horse. Although it may seem odd today to assign soldiers jobs based on their height, Napoleonic armies made it a policy to give different men different jobs. Smaller men were genuinely better-suited for the quick-moving, active work of a light cavalryman.
On the campaign trail, light cavalry were responsible for scouting, preventing enemy forces from scouting, interfering with enemy foraging parties, interdicting enemy supply lines, and the like. They'd often act as pickets, standing guard on the periphery of the army as it moved and camped. A moving army would usually be preceded and flanked by such small parties of light horsemen. On the battlefield, light cavalry skirmished on their own or supported skirmishing infantry. Light cavalry weren't expected to throw themselves into prepared enemy forces, a al The Charge of the Light Brigade. In general, light cavalry sought to fight on the periphery of the battle. They'd scrap with other light cavalry and charge fleeing or disorganised enemy troops. Indeed, pursuit was an especially important responsibility for light cavalry. By riding down routed enemy troops, they could destroy an enemy army and stop it from fighting another day.
Heavy Cavalry:
As the name suggests, heavy cavalry had the largest men and the biggest horses. Most heavy cavalry troopers carried a long, straight heavy sword. Although also useful for making cuts, the heavy cavalry sword was designed to maximise the effect of a cavalry charge. A trooper would extended his sword in front of him as he charged, using the momentum of charging man and beast to impale his opponent on his swordpoint. Common types of heavy cavalry include the armored Cuirassiers and the Carabiniers (French Carabiniers-à-Cheval were armored after 1810). Their body armor and helmets were by no means impervious to bullets, but they did offer good protection against saber cuts and bayonet thrusts.
Heavy cavalry were extremely expensive to mount, train, and equip. A single heavy cavalry horse required generations of special breeding and great amounts of fodder. They represented a commitment on a national scale. As a result, heavy cavalry were a precious asset that were not to be squandered on pickets or scouting work. Instead, they were to be committed at a critical moment in the battle. They might ve sent to attack advancing enemy troops threatening a key position. Or they might br called on to destroy a wavering enemy formation. Ideally, heavy cavalry would attack in large numbers to maximise their shock attack. We have several accounts of over 10,000 heavy (and some light) cavalry making en masse charges.
Dragoons:
These troopers occupy an interesting space in warfare of this period. Firstly, dragoons aren't technically cavalry, they're supposed to be mounted infantry. The idea being that they ride to battle, dismount and fight on foot with their carbines. Indeed, the name "dragoons" comes from the French dragon, a name of an early firearm used by proto-dragoons. Secondly, so many different armies created different dragoon units and used them in different ways, it can be hard to pin down the "typical dragoon." In most cases, dragoons simply became regular cavalry.
The British Army of the period had three types of dragoons: light dragoons, dragoons, and later, dragoon guards. British light dragoons were essentially light cavalrymen, and were used as such during the American Revolution and the Napoleonic Wars. British dragoons were basically heavy cavalry, and were used as such, rarely fighting dismounted. "Dragoon guards" were not the elitr formations their name suggests. Instead, they were cavalry units who had been redesignated "dragoons" as a cost-saving measure (dragoons got paid less) and were given the "guards" title as a consolation prize.
French dragoons were used in a more diverse way. During the Peninsular War, French dragoons stayed very busy fighting Spanish guerrillas. Their skills at fighting on horse and foot proved useful in the rough terrain of Spain. In fact, due to shortages of horses, some dragoon units were dismounted and fought entirely on foot. This does bring us to one weakness of the dragoon concept. Since dragoons had to learn to fight on horseback with saber and on foot with a carbine, they were never able to excel at either. Infantry were still better infantry and cavalry were often bettet cavalry. In a sense, dragoons were often the jacks of two trades, but the masters of none.
Lancers:
The Poles are arguably responsible for spreading the lancer craze throughout Europe in the early 19th century. Indeed, the classical lancer uniform of the period is clearly derived from Polish military fashion. Napoleon even had a detachment of Polish lancers in his Imperial Guard and regarded them as some of his best troops. Uhlans and chevau-légers were common names for lancer-type units in this time period (although "chevau-léger" was technically a term for other typea of light French cavalry, so not all chevau-légers were actually lancers).
Lancers occupy a unique role in the armies of this time period. Although generally mounted on small- to medium-sized horses, their battlefield and campaign role was often similar to that of heavy cavalry (although lancers were used for some scouting and screening work).
If you'd like to know more:
Each type of cavalry had a unique origin and role, though by the time of the Napoleonic Wars some types (Hussars and Cheaveux Legere) had evolved to the point where their battlefield roles were neary identical.
Dragoons had originated as a type of mounted infantry, intended to dismount to fight rather than fight from the saddle. For this reason they originally were equipped with horses of a significantly lower calibre than their fellows in the cavalry proper. By the time of the Napoleonic Wars however, they were classified as proper cavalry. The retention of their short muskets, or musketoons, meant that they were significantly more flexible than most of their fellows. Other cavalrymen were restricted to pistols and blades, while dragoons could still function as a dismounted force. In some armies, Britain, dragoon regiments were designated as either heavy or light according to their intended purpose. Light dragoons scouted, foraged for supplies, protected the line of march, dispersed light troops etc. Heavy dragoons were intended to act as shock troops, filling the gap left by Britains lack of Cuirrasiers and Chasseurs au cheval. Light dragoons lacked the prestige of the Hussar regiments, were invariably led to friction.
Hussars were patterned on the famous Hungarian light cavalry of the previous century and wore flamboyant uniforms patterned on the originals. As the most prestigious light cavalry of the age, hussars enjoyed a reputation of bravery, intelligence and dash as well as rapaciousness, drunkenness and foolhardiness.
Cheaveux legere were yet another type of light cavalrly, preforming much the same duties as the hussars and light dragoons but their less flamboyant uniforms meant they were cheaper to equip and their lower prestige meant it eas common to mount them on cheaper horseflesh, further saving cost. Despite their ubiquity, the flashy nature of the hussars has meant the cheaveux legere have been largely overshadowed.
Uhlans and Lancers, the very best lance armed cavalry came from Poland, whose territory during the Napolonic Wars was partitioned by Russia, Austro-Hungary and Prussia. These light cavalry were armed with lances and excelled at fighting other cavalrymen, their superior reach and mobility allowed them to inflict great mischief on their opponents. When Napoleon created his own lancer regiments, he raised his regiments from green recruits. He had their lancers shortened somewhat to improve the ability for his troopers to handle them and they proved invaluable. In an age before cartridge ammunition rain could render black powder weapons useless, and it wet conditions lancers could attack infantry formations with utter impunity.
Cuirrasiers were the premier heavy cavalry of the age, big men on big horses with iron breastplated they could deliver an devastating charge against foot soldiers or lightly equipped cavalrymen. Some Austro-Hungarian regiments wore half breast plates with only the front protected, reducing weight and the strain on man and mount. These units were very expensive to maintain and difficult to replace but could provide a decisive battlefield punch.
Chasseur au cheval were another branch of elite heavy cavalrymen. Originally they did not wear breastplates, when ordered to reequip with armour they protested. They did not want to lose their unique identity and become clones of the Cuirrasiers, in order to salve their feelings Napoleon ordered their breastplates be gilt with bronze and this gave their armour a unique golden appearance.