What would happen if a small asteroid hit our moon?
Would we be in deep trouble or not, depending on where it hit?
it’s unlikely to alter the orbit of the moon, but a glancing blow (or even a very hard perpendicular impact I suppose) could eject a considerable quantity of lunar material, some of which could fall to earth (although it would might just be little pieces so may not actually do much harm). It would be a big problem for the tiny aliens that live there in their tiny space shuttles though. 
Last I heard, the prevailing theory on the origin of tektites is that they’re lunar debris kicked up by collisions which makes its way to Earth.
I doubt, though, that there’d be enough material sent our way to pse any real danger to us. We’d see a nice fireworks show when it hit, and then the Moon would have one more crater.
If something did hit it out of it’s orbit and closer to earth, could the moon come crashing into us? 
Depends on how big and what trajectory “it” had. Are you talking practically or theoretically?
Way too broad an OP. You have to define “small” asteroid, for one thing. If the asteroid was one of those ~1 kilometer dinosaur killers then we’d probably get a nice fireworks show. With maybe some larger inconvenient pieces of “splash” debris hitting the Earth.
If something really large hit the moon to the point of shattering it, we’d have some problems. I suspect that the Earth’s shape would change slightly due to the altered gravitational stress, and this would likely reflect in some very nasty seismic activity. Loss of the lunar cycle would have some unknown effect on various Earth creatures in terms of their ability to navigate, know when to migrate, and so on. There would be long-term effects on the atmosphere without the moon’s gravity helping to thin it out. Some scientists think that Venus’ thick and toxic atmosphere is due to the lack of a moon. No one would know when Easter occurred anymore. On the bright side, we’d have pretty Saturn-like rings, and tide tables would be a cinch to calculate.
Could the moon come crashing into us? Only under the most unlikely of circumstances where the kinetic energy of the asteroid cancelled out the moon's angular momentum around the Earth exactly. Or altered the moon's orbit into something wildly erratic. A hit like that would more likely shatter the moon. Realistically, anything big enough and moving fast enough to do that kind of damage is just as likely to hit the Earth as the moon -- this is where the real worries are.A great book with this premise is Moonfall, by Jack McDevitt.
It’s one of my favorite “doomsday” stories; I’ve read it several times.
Sheri
Recent theories of tektite origin is that it’s impact melting of terrestrial sediments.
Well… a near miss can rip away the earth protective layer, and crack the moon in half, therby unleashing magic and mutants on the earth.
Fortunatly Thundar will keep the servers up, so that the post apocalyptic future will still have the SDMB.
On a June evening in 1178, five British monks saw the moon’s northeastern limb sprout a cloud of fire and hot coals. Their account was chronicled in the extensive records of another monk named Gervase in Canterbury, England. Today, meteor scholars have identified this event with the 125,000-megaton explosion that carved the crater known as Giordano Bruno at roughly 36° N, 102° E on the moon’s northeastern limb.
OK, so are we saying that the entire moon would be shaken by an impact like this to such an extent that it (the shaking) would be visible from Earth?
If so, would such a shaking (or one from a lesser impact) erase the footprints left behind by Apollo crews?
This is Jack Hartung’s theory, but I don’t think it’s widely accepted. The moon was low on the horizon when this observation was made, so I think some sort of atmospheric distortion is a much more likely explanation of what was seen, especially the “throbbed like a wounded snake bit”. There’s a book called The Green Flash by D. O’Connell which is full of pictures of the distortions the atmosphere can cause in the rising or setting sun, and the moon would be subject to similar effects at times. I used to know Hartung slightly back in the 70’s and pointed this out to him, but did he listen to me? Noooooo.
It is highly unlikely that a even a large impact would cause shaking visible from the Earth. Seismic effects from an impact could erase the footprints of the astronauts, depending on the size and location of the impact. (If a good sized meteorite hit the landing site and excavated a crater, that’d do it, too. )
Re: The 18 June 1178 observation:
I have to agree that it seems unlikely that the shaking of the moon would not be visible from earth, especially by the unaided eye.
But… Looking at the cite provided so kindly by Attrayant, it also says:
“the “flaming torch [of] hot coals and sparks” describes the molten ejecta”
I thought the moon is believed not to have any “molten” mantle such as the earth has, or at least none so close to the surface that a meteor/asteroid impact could cause it to be thrown out as such.
Perhaps the material thrown up could reflect light brightly so as to appear like sparks and embers from a stirred fire? It wouldn’t have to be spurting lava, would it?
(Here’s the part where I show that I am no astronomer:)
Then, the Chronicle itself also says that:
“Now there was a bright new moon, and as usual in that phase, its horns were tilted towards the east and suddenly the upper horn split in two. From the midpoint of this division a flaming torch sprang up…”
Am I mistaken in understanding this to mean that the impact site would have been on the side of the moon that faces the earth? And wouldn’t this then mean that the impacting object would have crossed inside the moon’s orbit of the earth? And also, that it would have crossed ithe orbit from the direction of the sun headed outward?
That’s true. However, the kinetic energy of the impact would be more than enough to melt both the asteroid and the surface of the Moon. The thermal energy released would be truly enormous, in the order of thousands of megatons. You would see a LOT of “Hot coals and sparks.”
A tardy nitpick: for a large impact, the angle doesn’t matter. Craters are not created by stuff being shoved aside or knocked about by the impactor.
When a large impactor strikes the surface, its considerable kinetic energy is converted to heat. This vaporizes the impactor and a goodly amount of the surface, causing a tremendous explosion.
So, don’t think rock hitting the ground . . . think bomb going off.