Tides in the ocean are caused by the gravitational force between Earth and the moon. There are also atmospheric tides.
Lunar gravity affects the density of the thermosphere, which is the largest layer of the atmosphere. This is also where many satellites and the International Space Station orbit Earth. This lunar-induced drag is small, but it has to be included in the models used to predict the satellites’ orbits. The moon also affects the pressure at Earth’s surface.
When the moon is overhead, its gravitation pull causes Earth’s atmosphere to bulge toward it. Since the pressure at the surface is determined by the amount of air above you, the pressure, or weight, of the atmosphere on that side of the planet increases.
The fact that air pressure changes were correlated to the position of the moon was first detected in 1847 using ground-based measurements. The change is very small and was detected using careful statistical analysis on a long data record. A correlation with temperature at the surface was found in 1932.
A relatively recent study by atmospheric scientists at the University of Washington found that rain is slightly lighter when the moon is higher in the sky. You wouldn’t notice this difference as the change is only about 1 percent of the total rainfall.
Which is also not enough to impact other aspects of weather. The scientists discovered this by careful statistical analysis of special observation made from satellites over a 15-year period, from 1998 to 2012.