Santos Dumont's first flight of a powered heavier than air plane was in 1906. The Wright Brothers were in 1903-1905.
People had been flying for many years before this. Balloons and gliders were well known and in widespread use. What no one had done before then was to fly a heavier than air plane in a controlled, sustained, flight. This is different from just 'hopping' where you only attain flight for a very short period of time. Some people discount the 1903 Wright Brothers flights as 'hopping' since the longest only 59 seconds and they had the advantage of strong winds. However, by 1904 they had flown in a circle for the first time in history and by 1905 they were flying for over half an hour at a time. That is clearly powered, controlled, flight.
Dumont's first flights in 1906 were much like the Wrights first flights. Short, straight flights that lasted for under a minute. However, by 1907 he had built a plane that was clearly capable of sustained controlled flight.
What the "Dumont First" proponents often allege is that because the Wright Brothers used a launch rail their flight does not count. Dumont's plane took off using it's own wheels. The Wrights did not add wheels until 1910.
Personally, I don't think it matters much who was first. They both basically developed their crafts on their own and each made contributions to the future of air travel and each was building on the backs of people who came before them.
Scientific American has a pretty good run down of the different claims for First Flight if you're looking for more info.
I am not a historian, but I study aeronautical engineering and we had a guest professor speak about the first powered flight recently, so I thought I could add something to the discussion. He claimed that the first motor-powered flight was done by "Gustav Weißkopf" or Gustave Whitehead, a German who emigrated to the US. He flew a powered machine as early as 1901, two years before the Wright Brothers.
There is a lot of controversy around this claim. The wright brothers website even has an extensive article around Whitehead [1]. There are no good photographs or documents of this flight, and most of the information is based on witness reports.
However, in 1989 an accurate (although equipped with modern engines) replica of the airframe performed successful flights. [2]
The Whitehead foundation also has some interesting arguments. For example, the Smithsonian has a special contract that forces them to accept the wright flyer as "first" flyer:
Should the United States not prominently display the airplane, display it without the agreed-upon identification, or identify another airplane as being capable of controlled and powered manned flight before December 17, 1903, the ownership of the airplane reverts to the Estate. [3]
So they have a great interest in not identifying any other "first" flyers.
The popular "Jane's all the Worlds Aircraft" recognized Whitehead's flight as of 2013. [4]
So in conclusion I cannot answer the question definitely, but I could maybe share an interesting perspective on another aviation pioneer. It definitely must have been a very exciting time. Only 15 years later, aircraft were already fighting in WW1.
I Hope this Post follows the rules. It's my first time posting here and english is not my native languag, so please excuse my mistakes
1: http://www.wright-brothers.org/History_Wing/History_of_the_Airplane/Who_Was_First/Gustav_Whitehead/Gustav_Whitehead.htm Gustav Albin Weisskopf
2: https://web.archive.org/web/20060618013302/http://flightjournal.com/articles/wff/wff7.asp
In a way it depends on your definition of "Airplane." Merriam Webster gives the following: "A powered heavier-than-air aircraft with fixed wings from which it derives most of its lift." Source
Going by that definition, the Wright Brothers almost certainly deserve credit for the first airplane. However, let's look more closely. One thing that differentiates the Wright Flyers from Santos-Dumont's 14 BIS is how they took off.
In what is generally regarded as the first powered, heavier-than-air flight on December 17, 1903 the original Wright Flyer took off from a rail used to keep it strait and stay on the dunes at Kitty Hawk. "The flyer moved down the rail as Wilbur steadied the wings. Just as Orville left the ground, John Daniels from the lifesaving station snapped the shutter on a preset camera, capturing the historic image of the airborne aircraft." NPS Most importantly though, it "rose into the air on its own power and landed on ground as high as that from which it had taken off." ASME
Despite having a much more powerful motor - 50 horsepower - (J. Aerosp. Technol. Manag., São José dos Campos, Vol.4, No 3, pp. 355-379, Jul.-Sep., 2012) than the original Wright Flyer - 12 horsepower - (Smithsonian) Santos-Dumont's plane only managed a flight shorter than the Wrights'. (J. Aerosp. Technol. Manag., São José dos Campos). This was in September 1906, after the patent of the original Wright Flyer had already been accepted. However, from what I can tell it accomplished this without a rail, as none of the sources I've read mention one.
By this point in time though the Wright brothers had already managed a flight of nearly 40 minutes, vastly dwarfing anything accomplished by the 14 BIS, whose flights were still measured in seconds and meters instead of miles and minutes like the Wrights'.
"Once again, on October 5, 1905, Wilbur flew until his fuel was exhausted, only this time he had enough to remain aloft for 39 minutes, 24 seconds, covering slightly over 24 miles, a distance longer than all of the previous 109 flights put together. Unlike anything else in the world, it could take off, climb into the air, fly for extended periods in any direction completely under the pilot's control and land in a safe, controlled manner. And it had shown that it could do all of this over and over again." [ASME] While this Flyer was launched with the assistance of a catapult, I think its drastic outperformance of every previous attempt at flight is significant.
So while Santos-Dumont's contributions to aviation are significant (and worthy of their own full post), he did not create the first heavier than air craft capable of attaining and sustaining flight under its own power, or the first one to be fully maneuverable. Both of these were accomplished by the Wright Brothers, and so in my view (admittedly an American one) they deserve credit as having invented the airplane.
This is my first attempt at an answer on this sub, so I hope it meets the (needfully stringent) requirements.
Santos-Dumont was an independently wealthy man whose hobby was aeronautics. His projects were largely funded by his family's coffee plantation in Brazil and by some money he received from winning engineering contests (mostly in aviation). He loved showing off his work and his primary motivation was receiving recognition for it.
The Wright Brothers were not particularly wealthy bicycle makers with an interest in aeronautics who eventually closed their company to work on it full time. They loved it, but their primary motivation was to make money off of their projects and the recognition they wanted was in the form of patents and government contracts. The Wrights were paranoid that their designs would be stolen so they intentionally did not fly in front of many people. When they began to receive recognition they refused to fly in front of anyone at all. They did not want pictures taken of their device and they developed it in remote areas, so there was limited evidence that it occurred.
During this time period when the Wright Brothers refused to fly, the Fédération Aéronautique Internationale was founded (two years after the Wright Brothers first flight) in Paris where Santos-Dumont lived. They put forth a set of rules for what they would consider to be the first flight. Those rules were:
A) the flight should be done before an official organization, qualified to ratify it; B) the flight should be done in calm weather and over a plain ground, and properly documented; C) the machine should be able to take off from a designated area by its own means with a man on board; D) the machine should carry on board the necessary source of energy; E) the machine should fly in a straight line; F) the machine should make a change of direction (turn and circle); G) the machine should return to the starting point.
At that time, the Wright Brothers weren't about to show their product to any official organization out of fear it would be stolen (I cannot emphasize this fear enough, they wouldn't even show it to potential buyers and demanded that they sign the contract sight unseen). They also preferred a catapult like system as it shortened the needed runway and made takeoff significantly safer. So Santos-Dumont was the first to have an acknowledged flight under those rules by that governing board which is why Brazilians believe that he invented the plane.
However, had they wanted to, the Wrights' plane was capable of liftoff without the rail system (but again their main goal was commercialization, so they were focused more on getting it off the ground safely and keeping it up there). And in 1908 when they finally got a government contract, they brought their plane to Paris where the FAI reluctantly agreed that the Wright Brothers knew what they were doing and it was likely that they had been the first. Their plane was the most technically advanced thing they had seen and flew farther, higher, and longer, than any other plane previously. Between the patent attempts going back to 1903, the fight for government contracts, and the superior engineering made it impossible to believe that they had just come up with this design in only two years.
ETA: I don't know if quora counts as a source, but it does the best job of explaining why the Wright design was superior where most other sources I've seen just quote it as "superior".
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In a similar vein, in New Zealand we are told that Richard Pearse was the first to fly. Is there any truth to that at all?