Experimental Units I am doing a paper airplane project for school.
I am doing a two-factor experiment, each with three levels, the factors are design of the plane and angle of launch. I am also having three people make a copy each of the airplane designs to account for variation when manufacturing the planes.
Am I correct in believing that there are 9 experimental units (3 designs $\times$ 3 people each making each design)? 
Or are there 81 experimental units (3 designs $\times$ 3 launch angles $\times$ 3 people making a plane each $\times$ 3 repetitions).

If your factors are design of the plane and angle of launch you have three of each so $9$ cases.  You have decided that the manufacturer and the repetition do not matter systematically, so repeating them will reduce the random variability.  You have $81$ data points that you have decided to bin into $9$ bins of $9$ points.  
This is a reasonable approach when you cannot afford many trials.  You have to guess what matters and what doesn't.  The problem comes when something you thought was uniform is not, like one manufacturer is better than the others.  Maybe s/he creases the paper better, so there is less air resistance to the flight.  Then the one of his/her planes that got launched at  a better angle will bias your selection toward that design.  
It is hard.
