What was the Order of the Dragon?

Counted among the members of the Order of the Dragon were Vlad II Dracul and Vlad Dracul III (known later as Vlad the Impaler, and the start of all the Dracula myths). 

But what did being a part of the Order of the Dragon mean? And who else was part of the Order? 

Founded in 1408 by  Sigismund, King of Hungary, the Order of the Dragon was a monarchical chivalric order (meaning they had to be a reigning monarch or former reigning monarch to join). The original 21 members were: 

Stephanus despoth, dominus Rasciae, item — Stefan Lazarević, known as "the Tall" (1374–19 July 1427); Serbian Prince (1389–1402) and Despot (1402–1427) 
  1. Hermannus comes Cily et Zagoriae, 
  1. comes Fredericus, filius eiusdem, 
  1. Nicolaus de gara, regni Hungariae palatinus, 
  1. Stiborius de Stiboricz alias vaiuoda Transyluanus, 
  1. Joannes filius Henrici de Thamassy et 
  1. Jacobus Laczk de Zantho, vaiuodae Transyluani, 
  1. Joannes de Maroth Machouiensis, 
  1. Pipo de Ozora Zewreniensis, bani; 
  1. Nicolaus de Zeech magister tauernicorum regalium, 
  1. comes Karolus de Corbauia, supremus thesaurarius regius, 
  1. Symon filius condam Konye bani de Zecheen, janitorum, 
  1. comes Joannes de Corbauia, dapiferorum, 
  1. Joannes filius Georgii de Alsaan pincernarum, 
  1. Petrus Cheh de Lewa aganzonum regalium magistri, 
  1. Nicolaus de Chak, alias vaiuoda Transyluanus, 
  1. Paulus Byssenus, alter Paulus de Peth, pridem Dalmatiae, Croatiae et totius Sclauoniae regnorum bani, 
  1. Michael, filius Salamonis de Nadasd comes siculorum regalium, 
  1. Petrus de Peren, alias siculorum nunc vero maramorossensis comes, 
  1. Emericus de eadem Pern secretarius cancellarius regius 
  1. et Joannes filius condam domini Nicolai de Gara palatini. 
(List is from The Order of The Dragon Wikipedia page) 

Although the original Order was to have 21 men, it was expanded in 1418 to include an extra 3, making the total 24. When this number was not enough to keep up with the demand to join, more inductees were let in between 1431 and 1437. 

With the introduction of these new recruits, a two tiered system was put in place.  

The new inductees, which included all new members who were not in the original 24, wore the emblem of the dragon and were referred to as Second Degree. 

The original 24 wore the cross and the dragon and were referred to as the Superior Class. 
With Vlad II Dracul and Vlad Dracul III not joining until sometime in the 1430s, this would make both the father and son Second Degree Dragons.  

There were many of these kinds of Orders. They were built upon romanticized stories of the Christian military orders at the time of the Crusades. 

Sigismund started the Order of The Dragon to defend Christianity from any enemy of the religion.
At the time, the largest enemy to Christianity was the Ottoman Turks. The Order of the Dragon was hugely popular in what would become Germany and Italy, but with the death of its founding member in 1437, the Order fell out of fashion in Western Europe. 

However, in 1453 after the fall of Constantinople, the Order of the Dragon would play a role in the
Ottoman wars in Romania. Vlad Dracul III at a young age, was a political prisoner of the Turks. With the constant punishment received, it was thought that it was this part of his life that turned him into the sadistic killer he was later known for. 

In 1457, Vlad helped his cousin to ascend to the throne of the Principality of Moldavia, driving out the Ottomans.  Vlad Dracul III would later on battle with the Ottomans to regain his throne of Wallachia twice.  

Does the Order of The Dragon still exist today? 

It could very well, and we are not aware if they have adopted a new name. Any search on the internet only shows the history of the Order but nothing that would link to it being a functioning group today. 

My best guess, and this is a guess, is that it is not around anymore. With wars with the Ottomans continuing well into the 1500s, allegiances change. The likelihood that the descendants of the founding families took up membership when they had their own battles to wage, would be a slim chance. It's all about the context of the times.

Any of the driving voices of the Order would be dead and gone by the time the 1500s started. 

Although the Order may be gone, the symbols of the Order of the Dragon live on. The cross and the dragon were adopted by many Hungarian noble families to use in their house crests.