Right off the top of me 'ead (I'll do some research later): I read somewhere that a Hzar was a company of at least 19 mounted men, under the command of a nobleman as a sub-unit of Hungarian light cavalry regiments. As time went on, this light cavalry tradition spread throughout European armies (it probably started with Attila's fast-riding hard men around the end of the Roman Empire). I think we copied the dandified French model. Lancers were just that - heavy horses, big riders - descended from mediaeval knights' traditional role. Dragoons, of course, were simply mounted infantry - forerunner of today's Mech Inf. I believe the Light Cav were originally designed as a reconnaissance/skirmishing force. Now - I'm off to do that research...
I think the use of the Lancer and Hussar type uniforms were primarily a fashion thing in the days when the Colonelship of a regiment was a business arrangement and the soldiers were dressed to demonstrate the Colonel's status . I don't believe that the function of the cavalry changed when their titles did just the dress and the titles. They were , irrespective of title, light cavalry while the regiments of Horse were heavy cavalry.
By the time British Hussars were formally raised in 1805, the distinctions between different brands of cavalry were becoming blurred. The Europeans had seen the Hungarian and Polish versions of Hussars since the 16th century, and the role had changed according to the nature of campaigns and the weapons developments of later years. Heavier armour in response to improved firearms, turned some mounted soldiers - nominally Hussars - into heavy cavalry. The Poles actually had "Light" and "Heavy" Hussars. But no campaign illustrated the differing roles better, than the Crimean War. On the same day, the British heavy cavalry won a brilliant victory - and the light cavalry dashed themselves to pieces against Russian artillery. Apart from dress traditions, cavalry units don't seem to be distinguished from each other by the roles they perform these days. They carry out either armoured reconnaissance or armoured assault, depending on whether they're in CVR(T)s or Challenger 2s.
Hussars were 'sexy' and had flamboyant uniform which was important in the 18/19th century when Armies were dressed to look good. All Hussar Regiments had a variation on the Hungarian hussar uniform. There were differences in regimental roles. Cavalry was 'heavy' or 'light' and the french put a dozen regiments into armour and called them currasiers as did the Austraians, Russians, Saxons, Polish and anyone who could afford to. They were kept en masse for decisive charges and were exempt from scouting, escort and outpost work which light cavalry did. Young bloods joined the hussars and were the wild things of the army. La Salle, the greatest leader in the French napoleonic cavalry often said that a hussar should be dead by 30! He lived to about 34 and died leading a french heavy cavalry charge (his horse being light outdistanced the heavy cavalry mounts and took him into the enemy forces ahead of the main body).
Towards the end of the napoleonic wars there was little difference between the regiments as attrition, lack of suitable horses and general war weariness affected the elan of the hussar, and the rest of the cavalry.
The French also influenced British army 'fashion' infantry of the line adopted a derivative of the French kepi in the 1860s. The HLI and Cameronians (Scottish Rifles) continued to wear that style of cap until 1914 (and until 1939 in levee dress), and by the Cameronians' Bugle Major until their disbandment in 1968. The cap was replaced among most line regiments, following the Franco-Prussian war, with a distinctly Prussian looking helmet which continues to be worn today by several English regiment's Corps of Drums.
The Zouaves were French, originally North African, and the Americans adopted them because of closer connections to France than to Britain, at the time (independence war, statue of liberty, and the British support of the South).
Hussars existed in the German and Austrian armies up to WW1.
Dragoons were traditionally heavy cavalry, whilst Hussars were little different than Light Dragoons. Several Light Dragoon regiments re-roled as Lancers following the Napoleonic wars.
Some colonial units adopted the Zouave-style uniform - notably the Gold Coast Regiment (Royal West African Frontier Force). The kepi remained in service with the RCT band until their absorption in to the RLC.
Get a copy of Light Dragoons by Alan Mallinson. Obviously focusing on 13th, 15th, 18th and 19th Hussars / Light Dragoons, it explains in detail where they originated, how frequently they changed their name to suit their styling (and sometimes didn't change their names to suit their stying).