You are prety much describing the most basic operation of a monome www.monome.org.
Of course the monome can also do a LOT more, but your project is where most people start.
The question though is can you do it more cheaply (I'm guessing - you didn't provide a budget, but lets face it, 99% of the time it's "as cheap as possible")
Since you can connect it to a PC, I'd just use a simple sketch with 16 buttons connected to the arduino. This leaves you enough pins for a serial connection back to the PC. On the PC side, you'd need to use software like processing,max/msp etc (I'm not an audio geek) to play the sounds corresponding to each button.
Ofcourse you could make it more complicated/neater by using input shift registers, but from your questions it sounds like the simplest would be the best way to go.