The effect that the explosion of the moon would have on Earth depends on the nature of the blast. If the moon were atomized, the result would be extremely different than if it broke into large chucks, which would fly off in every direction. The former event would have serious repercussions for the planet, in terms of gravity, average length of a day, and the tides, whereas the latter event would be the end of all life on Earth.
If the explosion of the moon caused it to simply disintegrate, with no sizable chunks remaining to hit the Earth, the effects on the planet would be entirely gravitational. The worst problem would be the lack of gravitational pull on the Earth, allowing the planet to revolve faster, resulting in shortened days. With only the sun's gravitational influence, the tidal ebb and flows would be slowed. The difference between high and low tides would be reduced, meaning the tidal drag, which slows the Earth down, would decrease, resulting in a six- to eight-hour day. Another problem would be that the planet’s center of gravity would change, adding more of a wobble to Earth's orbit around the sun. The Earth’s climate is affected by the 23.5 degree tilt of the Earth's axis. Without the stabilizing presence of the Moon, there would be dramatic climate changes.
On the other hand, a violent explosion of the Moon that created huge, hurtling debris chunks would inevitably cause the total extinction of all life on Earth. When the debris hit the surface of the Earth, the planet would be battered beyond recognition, and all civilizations would be obliterated.