As legend says, after they lost Battle of Kosovo (1389) Serb units, most notably their light cavalry, have spread to Hungary and then further over Europe.
In Medieval Hungary, these became known as hussars since about 1432 (Oxford Encyclopedia of Medieval Warfare and Military Technology, p. 306); they were greatly developed by Matthias Corvinus (1443-1490). At the time hussars differed from the heavy cavalry by lack of any armor; their only protection was an asymmetric shield (ibid).
Depiction by Tadeusz Korzon (1932).
To avoid a misunderstanding. Specifically in Polish-Lithuanian Commonwealth the hussars were further developed into heavy "winged" cavalry (1500-1770), and small numbers of similar heavy cavalry were employed in Russia. But the usage of term "hussars" as heavy cavalry is limited only to these countries. In Polish language even the word used for the heavy-cavalry-hussars ("husaria", "husarze") is different than the light-cavalry-hussar ("huzarzy").
For the rest of the Europe, the hussars were the light cavalry. Some Hungarian units became mercenary with other European armies since 1526, when Hungary began to be overridden by Ottomans. Other European hussar units simply mimicked the Hungarian ones. They retained their Balkan-styled uniforms (dolman jacket, tall fur cap), but with time dropped the shield and the long wooden lance in favor of firearms.
Differences from other flavors of light cavalry:
- lancers - just a general name for any troops using lances
- dragoons - French origin, historically fought more on foot
- uhlans - Tatar/Mongol origin, tactically similar
- cossacks - Cossack (Ukrainian) origin, tactically similar
Also, the English Wikipedia article that you use is of very very poor quality, so in this specific case I wouldn't use it as source.
UPDATE So why the hussars became so popular in respect to the other light cavalry units of each of the nations? I found no direct sources, but many indirect ones, to point to their sexual attractiveness to women caused by Balkan-style flamboyant dress, mustaches, hairdos (Swords Around A Throne pp. 240-241). This would be a major factor as we entered the times when men needed to be recruited, motivated, and in many cases served voluntarily (Hungarian Hussar, p. 8). There was no military advantage over other contemporary light cavalry units (Swords Around A Throne pp. 240-241).