After the Sun, the Moon is the brightest object in the sky. The phases of the moon are explained by the orbit of the moon around the Earth, and the times at which we see the moon are linked to its phase. The moon orbits the earth anticlockwise when looking down onto the north pole. ALWAYS DRAW IT THIS WAY AROUND! New moon is when the moon is lined up sun-moon-earth. It then rises and sets with the sun (ie rises at ~6am, is on the meridian at midday and sets at ~6pm. But the next night the moon has moved around a bit in its orbit around the earth. It goes anticlockwise so we the right hand side of the moon goes into sunlight first, and the moon rises later and sets later. At first its a (waxing) cresent in the evening sky, with righthand side bright. After ~7 days its got a quarter of the way around so we see the moon as a half moon (right hand side sunlit - first quarter) and its on the meridian at sunset, so it sets about midnight (it rose at around midday). keep on going and you get the moon rising progressively later and later (waxing gibbous) until it gets around to full moon when it lines up sun-earth-moon and it rises as the sun sets at ~6pm, is on the meridian at midnight and sets as the sun rises around ~6am. keep on going and now its the left hand side of the moon thats in sunlight (waning gibbous) and the moon rises later. After ~7 days its round to third quarter so we see the moon as a half moon with the left hand side bright. it rises ~midnight, and is on the meridian at ~6am (sunrise) and sets at ~midday. And so back to the beginning via a waning cresent (left hand side bright). There is a nice java amimation - the default point of view is top view, but its better to select 'both' and then hit animate.
The orbit of the Moon around the Earth is tilted slightly with respect to the plane of the orbit of the Earth around the Sun. To get an eclipse the planes of the orbits need to line up (which only happens twice per year) AND the moon phase has to be either new (for a solar eclipse) or full (for a lunar eclipse). It is a lovely coincidence that the moon and sun have the same angular diameter on the sky (the sun is much bigger than the moon, but it is also much further away), so the disk of the moon almost exactly covered the disk of the sun, allowing us to see the solar corona. There is a nice introduction to lunar eclipses with an animation, and solar eclipses with an animation.
The moons orbit is not very circular - it is an ellipse. So there are times when it is closer to the Earth than others. Only when the moon is closer than average to the Earth is its angular size big enough to completely cover the Sun. Solar eclipses at other times give an annular eclipse
The moon rotates! But its rotation period is exactly the same as its orbital period around the Earth, so we always see the same face of the moon towards the Earth. More on moon rotation. This is NOT a coincidence. It is due to the gravitational interaction between the Earth and the Moon. The effect of the moons gravity on the Earth causes tides in the sea and also causes (much smaller!) tides in the rocky crust of the Earth. These will eventually bring the Earth into co-rotation with the moon. The effect of the Earths gravity on the moon is rather larger, since the Earth is larger so has more gravity. But the moon has no liquid seas, so the Earth produces tides only in the rocky crust of the moon, and the interaction of these bulges with the Earths gravitational field slowed the moons rotation, locking it with the orbit.