1. As a symbol, the moon represents feminine power, but also the rhythm of cyclic time, and universal becoming. It's a symbol for the unseen aspect of nature. It represents the irrational, the intuitive, and emotions. The moon is a bringer of change.
2. The Moon has always been shrouded in mysteries. All the missions we have sent to learn more about her, have only raised more questions than they answered, as they revealed an increasing number of anomalies. Rocks, e.g., were found on the Moon's surface that are more than half a billion years older than our solar system, and that are not the remains of a meteor impact. Unlike Earth, the Moon hardly contains any iron, while so-called refractory compounds like Titanium, that are heavy in nature and therefore should sink below the surface, are found on its surface, whereas on Earth these compounds did sink below the surface. Thus, both the age and the composition of the Moon indicate that she is not originally from our solar system.
Then there is the problem of the Moon's orbit. It is quite unlike the orbits of other bodies in our solar system: those are elliptic, where the Moon's orbit draws an almost perfect circle. The fact that the same side of the Moon always faces Earth is another anomaly. But even more difficult to explain is how the Moon got here, in an orbit around Earth, in the first place, as the Earth's mass and gravitational pull aren't sufficient to capture it. It may sound hard to believe, but hitherto, no satisfactory scientific explanation has been given.
To add to the mystery, there are the numerous tales that speak of peoples and times before the Earth had a moon. There are, e.g., Sumerian, Greek, Roman, Finnish, Tibetan, and Latin American texts that all state that the moon's arrival and orbit around the Earth are fairly recent.
Then there are the accounts of the lights and shadows that have been witnessed, and the artefacts that have been found: there are observations of tracks in the moon dust; of bridges that appeared virtually overnight; and of structures that cannot be explained by any known natural process. The moon even seems to have its own pyramids! (The so-called Brookings Report explicitly refers to the artefacts that have been found).
Conspiracy literature mentions the existence of several extraterrestrial and mixed extraterrestrial and human underground bases on the Moon. Most of these would have been built by secret organizations that consist of both alien and human members. There are even theories that the US Government would have been involved plans to start a colony on the Moon as a possible evacuation destination should a nuclear war on Earth threaten life on the planet. These plans are often referred to as "Alternative 3." Many sources include Mars as a possible destination, too.
3. In the Tarot Decks, the Moon is card XVIII of the Major Arcana.
4. Astrology: TBA.