The most immediate effect (other than the lack of moonlight, of course) would be on the Earth’s tides. With only the Sun’s gravitational influence, the difference between high and low tides would be reduced dramatically – as would tidal drag, which slows the Earth down at a rate adding about 0.002 seconds to the length of a day each century. Long term, the effects would be far more serious. The climate of the Earth is sensitively dependent on the 23.5° tilt of the Earth’s axis, and without the stabilising presence of our relatively huge Moon, the gravity of the other planets would produce big changes in this angle – as it does with Mars, whose tilt changes by 60° over a few million years.
It depends on the manner of the moon’s destruction. If it was, say, zapped to bits by a Death Star and those bits still floated in a cluster in the same orbit, I expect they would exert the same gravitational pull on Earth as does the intact moon, and not much would change on Earth.
We’d no longer watch the phases of the moon at night, but see a glittering cloud of debris which would probably be a lot brighter than the full moon, what with all those zillions of little surfaces to reflect sunlight. I know some astronomers who would really hate this new interference with their dark skies.