Short answer: Technically it's possible that the Earth and Moon could collide in the very distant future, but it's very unlikely. It's certainly not going to happen while any of us are alive.
Long answer: The Moon is in a stable orbit around Earth. There is no chance that it could just change its orbit and crash into Earth without something else really massive coming along and changing the situation. The Moon is actually moving away from Earth at the rate of a few centimetres per year.
In theory, if another very massive object such as an enormous asteroid came close enough to the Earth/Moon system, it could possibly disrupt things enough for a collision to happen. However, we have a good understanding of where things are in our part of the Solar System and it's extremely unlikely that anything this massive could come so close to us.
In the very long-term future, things are less clear. It's possible that something could come from the outer Solar System, or even from outside our Solar System. This is unlikely though.
Another possibility is that the Moon could eventually stop moving away from Earth and start getting closer. The good news is that this wouldn't happen for billions of years, after the Earth and Moon become tidally locked with each other. By that time, the Sun will have expanded to become a red giant and Earth probably won't be habitable anyway.
What would happen if the Moon crashed into Earth?
Everything on Earth would die. The only way to survive this collision would be to leave Earth.
The Moon and Earth would both be destroyed; the Earth would probably be split into numerous smaller pieces. Game over.
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