As Leo mentioned, all aspects in the geocentric system of astrology are based upon the the angles the planets appear to make to each other from our perspective here on Earth.
The last example you gave of the Moon square the Sun is probably the easiest to understand, since the phases of the Moon correspond with its relative position to the Sun as it revolves around the Earth. The New Moon corresponds to the Sun-Moon conjunction; the First Quarter Moon corresponds to the Sun-Moon square (with the Moon 90 degree ahead of the Sun in the zodiac); the Full Moon corresponds to the Sun-Moon opposition; and the Last Quarter Moon corresponds to the Sun-Moon square (with the Moon 90 degrees behind the Sun in the zodiac). The Moon forms other aspects to the Sun, of course, such as the trine and the sextile, but the dynamic aspects are the easiest to see.
During the New Moon phase, the Moon is still reflecting light from the Sun although it does not appear to be doing so from our perspective. This is confusing at first when you consider the aspect that corresponds with the New Moon phase--the conjunction of the Sun and the Moon; it may be one's natural tendency to assume that this would be the aspect in which the most light from the Sun could be seen. Yet the Sun does not revolve around the Earth, as early astronomers once thought based upon the apparent movement of the Sun rising and setting in the sky, and although one side of the Moon certainly reflects light during the New Moon phase, it is not the side of the Moon that we can see. So during the opposite Full Moon phase, when the Moon is opposite the Sun, we can see the entire surface of the Moon because the Moon has moved into a position in which the side the faces the Earth reflects all of the light of the Sun.
Does my description make any sense, or is this Mercury Retrograde period completely addling my brain?
A picture can probably explain this phenomenon much better than any words: