Aerodynamics was a fledgling field in 1900-1903 and the Wright brothers made some very significant contributions to it. There was some preexisting data and models of lift but it was either poorly understood or just wrong (http://wrightstories.com/lilienthal-data-not-in-error/). Fed up with issues with this data, the Wright brothers built a wind tunnel with the capability of measuring both lift and drag (http://wright.nasa.gov/airplane/tunnel.html). Their propellers were also a great technological advance. There was only data on props and screws being used in water and they couldn't put a prop in their wind tunnel so they had to develop them using theory. Amazingly, they created a stunningly efficient prop (http://www.uh.edu/engines/epi1867.htm), as later computer simulations would show. They were truly the leading aeronautical engineers of the day.

were their engineering theories, mechanical ideas, etc... sound enough to help influence and drive the huge leaps in aviation technology that occurred early on

Not just early on, but to this day. Every airplane relies on controlling differential lift (giving one wing more lift than the other) to allow it to turn ("bank" technically, but the result is turning). Even if you only wanted to fly in a straight line you'd still need that ability in order to keep making corrections so that it continues in a straight line (much like how even if you only wanted to drive in a straight line you'd still need a steering wheel).

The Wright Bros. used "wing warping" to accomplish this, where they literally twisted the wings to give one more lift than the other. That didn't work so well as planes got bigger and more rigid, so they switched to moveable control surfaces called ailerons. The principle is exactly the same though.

Of the Wright Bros. eight patents, seven are for controlling an airplane in flight (the eigth is for a toy). They made innovations other than diffential lift, but that was the big one. They weren't the first to make wings shaped well enough to give decent lift, or to bolt on an engine an propellor. What they added was the final piece of the puzzle - controllability. Without that a plane will quickly crash, which is what happened with earlier attempts to make an airplane.

Many of the responses to this thread are about control surfaces. But I will talk about some of their airplanes. The Wright Brothers convinced the US army that airplanes were the way of the future, and that it was something to look into. So the Army sent out an order for an airplane that could:

-Fly for an hour

-Carry both a pilot and a passenger

-Attain a speed of 40mph

-Land 10 miles away undamaged

The contract was awarded to the Wright's Model A design. The brothers were paid $30,000 for their invention. If you count inflation then that money is now worth $787,000.

They continued to build airplanes for the beginnings of the Army Air Corps until their Model C infamously developed the bonus of uncontrollable nose dives.

Anyways the Wrights built not only the first airplanes, but the first military airplanes for both the Army and the Navy. They made airplanes for the Navy by placing pontoons at the bottom of their regular designs but I believe they were unsuccessful.

It is important to note that they are not simply the first who got off the ground.

There were balloonists who had made multiple flights experimenting with both hot air and hydrogen as far back as 1783.