For approximately 4.5 billion years, the moon has orbited the Earth, stabilizing the tilt of its axis. Without the presence of a moon, the tilt could vary by up to 10 degrees, causing climate change on a global scale. Fortunately, Earth does have a moon, and such concerns are largely irrelevant.
A much more plausible effect is the moon’s influence on the tides. The moon’s gravitational force pulls the water towards it, raising the water level on Earth’s surface. Since gravity becomes weaker with increased distance, the water directly below the moon is pulled more strongly than the water to the side. This makes the ocean bulge, causing the sublunar tide. By similar logic, the water on the opposite side of the Earth experiences the weakest force because it is the farthest away. Since the water there is pulled weakly compared to the water on the sides, the water on the opposite side is actually pushed outward, creating the antipodal tide.
A surprising aspect of the tides is that they do not occur exactly below the moon - the Earth’s rotation keeps them slightly ahead. This disparity has consequences for the moon’s motion. Since there is additional mass in front of the moon, a force pulls the moon forward. One might observe that the bulge on the opposite side will exert a force in the opposite direction, but with a smaller magnitude since the other bulge is farther away. So the net force pulls the moon in the direction of Earth’s rotation, and Earth feels a corresponding pull in the opposite direction. Technically, these forces are actually torques.
So why does this torque matter? Well, since the moon is much closer to the Earth than any other sizable object, the Earth-moon orbit can be considered a closed system. With negligible external torques, angular momentum is conserved. Simply put, if something starts to spin slower, something else must spin faster. The Earth slows down because the moon pulls back on the bulge, so the moon speeds up because the bulge pulls it forward. Technically, the term is gaining or losing angular momentum. The Earth’s loss of angular momentum causes each Earth day to increase by about 23 microseconds (2.3*10-5s) every year. The corresponding gain in angular momentum for the moon causes the radius of its orbit to increase an average of about 38 mm per year across its elliptical path. That’s the same order of magnitude as the speed of your fingernail growth.
This change in the moon’s orbit - its precession - is slow, even slower than tectonic plate movements. But eventually, it would have some interesting effects. Currently, the moon is tidally locked to the Earth; that is the same side of the moon is always facing the Earth. Eventually, however, the Earth will also become tidally locked with the moon, so that the same side of the Earth always faces the moon. At that point, the Earth and the moon will orbit in sync with a common period of 47 Earth days.
Becoming tidally locked to the moon will have drastic effects for life on Earth. The pull of the moon will attract much of the water to a specific side of Earth, so there will be a permanent, unmoving tide. Additionally, the concentrated gravity of the moon could be capable of deforming the structure of Earth itself, changing its roughly spherical shape. Fortunately, we need not worry about such an occurrence. It will take 50 billion years for the Earth to become tidally locked to the moon. Earth has only existed for 4.5 billion years so far. Perhaps more importantly, our sun’s main sequence will only last for another 6 billion years. At that point, all its hydrogen will have been consumed, and the sun will expand dramatically, probably consuming Earth. But 2.5 billion years before that, the sun’s luminosity will have increased enough to sterilize all life on Earth anyway. So while the precession of the moon is a fascinating idea, most of us can live our lives without fear for its effects.