>>10569180We would be fucked beyond repair. If you're talking about another moon similar in size to the one we already have, that is.
Earth's gravitational field that has any significant influence on bodies only extends roughly 5 times or so the distance of the moon, or roughly 1-1.25 million miles. If you placed a Luna sized moon between Earth and Luna, the resulting gravitational and tidal forces would either throw us out of whack, destroy the planet and moon, or some other terrible thing like messing up our weather systems and biosphere beyond repair.
If you placed a Luna sized object beyond the moon but still in earth's gravitational field, I don't know, but you still have a pretty bad time of it, all said and done. The Moon2 would probably fling Moon1 into a severe elliptical orbit around Earth, or slam it into us.
That's not to say of course that Earth can't have another orbiting body around it. We could maybe have a moon below or up to the size and mass of something like Pluto, or Neptune's moon Triton without anything too terrible happening. But they would have to remain on the out side of our gravitational field if we want them to stay stable. Maybe even occupy a Lagrange point and only interact with the Earth-Moon System every couple of months- that way it'd still be a "moon" in the sense that it's affected by the chief gravitational body (Earth) occupying the same orbit, but it wouldn't be so uncomfortably close to us all the time.