Project A119 was a plan to launch an intercontinental ballistic missile at the Terminator, which is what John Connor would have done if "teaching it how to love" didn't play out in T2. Wait, sorry--the "Terminator" in this case refers to the dividing line between the dark and light side of the moon.
A large explosion on the Terminator line would put the sun behind the mushroom cloud, making the explosion visible with the naked eye from Earth. Presumably this is because the scientists in question were using prison logic: If you don't want to end up somebody's man-wife, you gotta kill the first random guy you see and make everybody else think you're crazy.
"You wanna survive in space? You gotta nuke something on the first day or else become Saturn's b***h."
The entire project was classified, of course, so it was given a slightly more innocuous sounding name than Project Nuke-A-Moon. They decided to call it "A Study of Lunar Research Flights," which may have been going a bit too far; that's like calling a serial rapist a "Can-do Casanova."
Fortunately, the United States came down off their explosion high and realized that nuking an orbiting planetary body for no particular reason might cross the line between "illustrating our technical prowess" and "cartoonish supervillainy," so they scrapped the plan and sent Armstrong up to land on the bastard instead. Think about that for a second: Neil Armstrong is considered a valid replacement for a nuclear missile.
"The name's Armstrong, but friends call me A-Bomb."