I'm no orbital dynamics expert, but the Moon's period of rotation around Earth is 27.3 days, roughly, and it rises 50 minutes later each night than the night before, so eventually, it will be somewhere overhead during daylight. The Moon has four major phases: New Moon, when it is roughly between Earth and the Sun, so the backside is lit and we can't see it; 1st Quarter, when it is half lit, and the lit side faces West; Full Moon, when the relative position is opposite the Sun from Earth and the near side is fully lit; and 3rd Quarter, when it is again half lit, but the lit side faces East. If the phases are moving from New to Full, it is a waxing Moon, if moving from Full to New it is a waning Moon. The period between New and First Quarter, and between Last Quarter and New is a crescent; the period between First and Last Quarter, on either side of Full, is a Gibbous Moon; you therefore will have waxing and waning crescents, and waxing and waning gibbous.
The Moon's orbit is 5.14 degrees above the ecliptic plane (the imaginary line where the planets' orbits are around the Sun), and the ecliptic is inclined 23 degrees to Earth's equator. This gives the Moon a varying "height" in its monthly orbit of Earth, so at times it appears higher or lower in the sky depending on what part of its period it is in. This difference in angle also accounts for why you don't have a lunar eclipse once a month in any given part of the World. The orbital plane of the Moon has to coincide with the ecliptic at New Moon, for an eclipse.
If you look closely during the moon's phases, you may also notice the exact positioning of features on the Moon will shift somewhat. This is due to what is called "albedo", which is a 'wobble' in the moon's presentation of its face to Earth. The Moon rotates once on its axis with every full orbit around Earth, so it always presents the same face to Earth; with the albedo it means that we can see about 60% of the moon's surface from Earth throughout its orbit.
I'm an amateur (with emphasis on 'amateur') astronomer, and look at the Moon from time to time through my telescopes, depending on weather and phase. It's best to observe along the terminator, the line between daylight and dark, because there are lots of shadows that emphasize surface features. So you have to observe different areas at different times of the month, and lately it's been difficult with all the nasty weather we've had. Full Moon has no shadows, and is way too bright unless you use a light reduction filter.
