It seriously depends how you define it.
Some might say Curtis's early planes because they had ailerons whereas wing warping was on the Wrights, but Curtis's control system was odd.
First semi-monocoque airplane would be significant from a structural standpoint, that being the Dornier D.1. This construction method is still used today, even on composite aircraft such as the 787.
Speaking of composites, those are becoming fairly popular now, and pretty much any airplane today has some composites (even though Cessna jets are aluminum, some fairing panels and control surfaces are composite, for example). By this argument, the DeHavilland Mosquito stands out for it's balsa-ply-balsa sandwich construction.
Instrumentation is important too, so first plane with a six pack. But then again, the first plane with a glass cockpit would be significant too. Or first with a HUD...the list goes on.
I think the true key is to find the balance, but there isn't really. I of course turn to the DC-3. It's aluminum construction, retractable gear, enclosed cockpit, etc were all very modern, but most of all it ushered in the age of aviation and airlines as a viable business, more than just a mail carrier that happens to carry passengers. The DC-3 made it's operators money by carrying passengers alone. Without it, no modern airlines would have ever existed.
Of course, that doesn't cover military or GA.
Finally, I'll finish with this thought: Aviation technology is constantly changing, and even technologies that aren't directly affiliated with aviation tend to get their first shot in airplanes. So defining a modern airplane is impossible. Would you consider a 737 modern? The design is 40 years old. Same story for a Citation Jet. The Cessna 172 is around 60 years old, and yet I would still look at parts of it as modern (the G1000 is amazing

). It's easier to define a modern car: 3 pedals (or two) always in the same location, a steering wheel thats normally the same size, the same set of gauges, doors, seats, engine, etc. But airplanes? Some have instruments from one manufacturer while others have them from another. Different planes for the same mission have entirely different configurations, and sometimes even different types of engines. Some are metal, some are plastic, high wing low wing, pusher puller, etc...There isn't one set standard that all others adhere to