“What is projection?” is certainly a good way to begin the thread, but a pertinent follow-up might be “How important is projection?”
Projection was extremely important for the 20th century Classical Concert Guitarist. Especially during the early part of the century, amplification was pretty crude. Andres Segovia, for example, went well into mid-century playing without amplification in Concert Halls simply because he thought the amplified tone, even through a microphone, was simply horrible compared to the natural acoustic sound of the guitar. A common joke in those days was “I went to hear Segovia last night”; reply: “You mean you really heard him?”
With those sorts of conditions, guitarists and luthiers did everything possible to add projection to the Classical Guitar. Certain wood combinations were found to help with projection, and new construction such as lattice bracing helped with projection as well. What some of those results also yielded, however, was a tone many found to be harsh; shrill to the point of being unlistenable.
Today, amplification is much more accurate than it was when projection was so important. Segovia himself relented and used a microphone during his later years. For that reason the emphasis on projection has faded. It’s not that an instrument that projects well isn’t still desirable. We feel it is, but we also feel its importance is now secondary. There just aren’t that many situations where you need Concert Hall projection from an acoustic instrument today, and of course on an Ukulele it was even trickier. Good tone today can be amplified accurately and heard under any condition, so if there is a choice to be made between tone and projection, most builders today will lean toward tone.