Dragoons were mounted infantry, at least originally. They used their horses as mobility to deploy rapidly, but they would fight in an infantry line or as light infantry. They were armed like most infantry and mostly fought as infantry.

Hussars, derived from the Serbian gusar, were a sort of light cavalryman, who were usually used in reconnaissance and raiding, typically armed with sabres, and later pistols (for use in the "caracole" tactic, where horsemen would move up to an infantry unit, empty their pistols, and then move to the rear to reload. As you can imagine, it wasn't very effective.) However, the Polish developed heavy hussars (the "winged hussars" of fame) whom were deployed like lancers.

Lancers were heavy cavalry that relied on melee shock action with the lance. That's it. The lance gave them maximal range, which was useful when dealing with spear formations, as well as taking advantage of the speed of the charge.

You also had cuiriassers, whom were armed with pistols and sword, and were notable for their use of the cuirass, giving them a heavier cavalry role. You may also hear "Uhlans." Uhlans were Polish cavalry in various Austrian, French, German, and Russian units.

I'm not too sure about Cavalier though, I always had interpreted to be another word for cavalryman.

Let's consider "back in the day" to be a period from roughly 1700-1880. I'm more familiar with the French and British armies of the period, but I'll try to discuss German, Polish, and Russian practice as well.

The basic divide in this era is between light cavalry and heavy cavalry. Uhlans/lancers, Chasseurs à cheval, hussars, and light dragoons are light cavalry. They're mounted on light, fast horses and are responsible for scouting and patrolling work. They also act as an ultra-mobile force on the battlefield. Some dragoons, cuirassiers, and carabiniers are heavy cavalry. Their job is to act as a sort of wrecking ball, smashing enemy units through brute force. Not something you want to see coming at you.

Lancers and Uhlans: Lances were initially something of an eastern European weapon, at least by the late 1700s. The Polish were especially fond of using lances and Polish lancers were probably the best in Europe during the 1700s and 1800s. One of the two lancer regiments in Napoleon's Imperial Guard was composed almost entirely of Poles. Lancers, like their name suggested, carried 2-4m long lances. Interestingly enough, not every "lancer" actually carried a lance. In some French units, only the front rank of horsemen carried a lance. The rest carried sabres, since if they carried lances they might accidentally spear the man riding in front of them.

After the Napoleonic War, the British Army raised several lance-armed units. Even after the Napoleonic wars, lancers saw action. The 17th Lancers participated in the Charge of the Light Brigade. The 21st Lancers (and a young Winston Churchill) charged Mahdists during Battle of Omdurman. And at the outbreak of war in 1914, there were still lance-armed German units.

Hussars: Hussars usually saw themselves as dashing, brave, excellent swordsmen and popular with the ladies. Think fighter pilots with horses and fur hats. In an era of fancy uniforms, hussars outdid them all. Seriously, take a look at this guy or this one

Hussars usually carried a short curved sabre. Some also carried pistols and carbines. As light cavalry one of their main responsibilities was to act as scouts and piquets screening the main army. In battle, they used their light, fast horse to maneuver around the flanks of the enemy and harass retreating enemy forces.

Cuirassiers: Big men on even bigger horses. They carried long, straight swords designed for thrusting. The last of Europe's armored cavalry, they wore helmets and breastplates. These didn't offer much protection from a musket ball but they could easily stop a sword or a lance. Heavy cavalry acted as shock troops, designed to use their weigh and momentum to punch holes in enemy lines. French Carabiniers also fit this this role. In Napoleon's army at least, cuirassiers and carabiners more or less served the same function. The British are basically the only Napoleonic army not to have armored cavalry, although they did use them during the War of the Spanish Succession and the Life Guards do wear ceremonial armor.

Dragoons: Dragons first emerge in the mid-1600s as mounted infantrymen, usually with a dragon (a short musket, and the reason dragoons got their name). Initially, they weren't intended for do much fighting on horseback. Instead they were supposed to ride to the battlefield, then dismount to fight from there. As time went on, they became more and more like ordinary cavalrymen. While they still carried carbines and trained to fight as infantry, Napoleonic dragoons did much of their fighting on horseback. Indeed, at least in the British Army by 1815, "dragoon" units were dragoons in name only and simply fought as regular heavy cavalry.

Their mobility and flexibility made them useful anti-partisan troops and the French used dragoons widely during the Peninsular War against Spanish guerrillas.

It would obviously differ a lot based on time period and which army they were in, but more often than not the basic distinction would be: Heavy Cavalry and Light cavalry, with many grey areas in between. Names would also imply a regional heritage for that style, Hussars being from Eastern Europe.

I am far from an expert but I can give you a quick overview on some Cavalry types from the Napoleonic Era:

Dragoons were mostly light cavalry, used for scouting and harassing. They were in essence mounted light infantry as they often carried carbines that, I believe, were for use on foot rather than in the saddle. They would have pistols and a Sabre, Curved sword, for use in the saddle, and when fighting in the saddle it was either to pursue an enemy or to counter enemy cavalry. They were most often used to run up to the enemy, take a pot shot on foot, and run away. There was a similar cavalry type, Carabineers who used a specialised carbine designed for use in the saddle, whereas the Dragoon's could, but they had a longer barrel and were more unwieldy. Both of these types would have been light cavalry.

Lancers were much like Dragoons, but they carried lances, and were pretty much only used by the french at this time. They were lightly armored, if any, and were generally used as the spearhead of a charge, trying to break the enemies infantry ranks. I believe they were fairly crap overall against enemy horse, as the lance is a crappy weapon in close quarters, so they would only go in if they would not be countered, or they would use a sword when fighting other cavalry.

Hussars were originally 14-15th century eastern european horsemen, and they continued to be used as light cavalry until World War I. I am unsure of their exact weaponry, but it would not differ much from the Dragoons, at this point being a Hussar was kind of a step above in status, but held little practical difference as far as I know.

Unsure what Cavaliers would have been in this era, I think they were more medieval.

Another unique french Cavalry were Cuirasseurs, which was one of the few remaining units that wore armor, They were heavy cavalry with breastplates, open helmets and some other miscelanneous pieces of metal, straight long swords and were designed to wreck infantry. They would charge straight at em, either making them run or die trying.

General heavy cavalry was only differentiated at this time with light cavalry by generally having broader, longer swords, heavier mounts and larger riders. The metal the french Cuirasseurs wore was little real protection against cannons or muskets, and plus the riders would not be the targets, the horses were. Much easier to hit. What they maintained from earlier years was the intimidation factor that size leant them. If the enemy infantry held firm, there was little cavalry could do solo, but if you made them run before you touched them, you already won.

Those are all the types that I can think off the top of my head for this era, and as you can see the trend towards indistinguishability (word?) was already well under way at this point, mostly in regards to heavy cavalry.

Does anybody know why that by the Napoleonic Wars heavy cavalry weren't equipped with lances? Cuirassiers specifically? Not even their first rank?

Given the main role of heavy cavalry and cuirassiers to act as a battering ram to break formations, I would think that the lance would be much more effective on the initial charge than the straight sword, even if it's immediately disregarded afterwards, but it seems that by this period the only use of lances were for skirmishing & pursuing retreats.