In the last 7 parts of "First to What?", I've told the story of the Wright Brothers, how they first developed the technology needed for controlled flight in a heavier-than-air aircraft, and then engineered and flew the first true airplane. I think it's fair to debate whether that was the 1902 glider, the 1903 Flyer, or the 1905 Flyer III.
Before they ever built their first prototype, the Wright brothers determined that wing warping in a banked turn (like a bicycle rider or a bird) was the way to make a controlled turn in the air, and soon found after the failure of their 1901 glider that the Lilienthal lift tables were in error, due to the use of the wrong value for Smeatons constant. This was at a time when others simply used Lilienthal's tables at face value, and tried to turn in the air like a boat on water in a flat, skidding turn. The 1902 glider was the summation of these insights.
The 1903 Flyer added an engine, but the Wrights had to design and build their own, and solve the problem of an efficient propeller. Their insight that a propeller isn't just an "airscrew" - it's actually a wing that moves through the air in a spiral, and the "thrust" is actually horizontal "lift". This allowed the Wrights to fly using just 12 HP, and combined with the 1902 controls, the 1903 Flyer makes a strong case.
But the 1903 (and 1904) Flyer was supremely unstable and difficult to fly. It wasn't until the 1905 Flyer III that the airplane was capable for sustained flight, and all subsequent Wright airplanes were derivatives of the 1905 design.
Were they first? Some are claiming that other pioneers such as John Montgomery in Seattle and Gustave Whitehead in Bridgeport CT, yet there is no documentary evidence that they ever got off the ground, but even if they did they never solved the problems of control - which in turn makes in unlikely that they actually did what is claimed. French pioneers like Santos-Dumont, Farman and Bleriot certainly did leave terra-firma, as is documented by contemporary reports and photographs, but they flew in uncontrollable craft in a more or less straight line and were lucky to survive. However the French "aviators" left their mark in the many French words we use - "fuselage", "aileron", "aviation" and "aeroplane".
For years the Smithsonian Institute claimed that Samuel Langley was the first to develop an airplane (the 1903 "Aerodrome"), and in fact the Langley "Aerodrome" was later successfully flown by Glenn Curtis - but hushed up was the fact that Curtis made many changes to aircraft to make it controllable, and put in a better engine and propeller - based on the Wright's principles. It was because of this dispute that the 1903 Flyer was dispatched to London's Science Museum until after WW2, it is on loan to the Smithsonian only as long as the Institute agrees to honor the Wright's achievement (which they do now very well).
Glenn Curtiss is probably came closest. By 1908, working with telephone inventor Alexander Bell, he has independently invented the hinged aileron and could fly over 1 mile (in a straight line). The next year he could also fly in circles and figure 8's. He took more risks and won more prizes than the Wrights, and ultimately bought their company winning the commercial and legal battles once Wilbur died and Orville lost interest.
Strangely, the wealthiest and most influential country of that time, England, did almost nothing of note during this period - except for a certain Winston S. Churchill, who took flying lessons in the period before WW1. His air-mindedness perhaps saved his country 20 years later, as he sounded the alarm over the Royal Air Force's unpreparedness to face the German Luftwaffe. The result was the RAF's beefing-up just in time to save Britain and perhaps the rest of the world in 1940.
So what exactly were the Wright's first to do? Not to fly, several had done so earlier than they. What no-one else did before them was to fully understand all the elements that make up successful, sustained and controlled flight, and engineer them together to make the world's first, true airplane. No aircraft built before the Wrights 1900-1908 development process did that, and afterwards no (successful) airplane built failed to use them. The Wrights were the world's first successful aeronautical engineers.