Months were originally based on the moon cycle — new moon to new moon — i.e. 29.53 days (ranging during the year between about 29 days 7 hours and 29 days 17 hours). Click here for NASA's article on moon phases, with a picture of the current phase as of today's date.
This cycle is due to the moon rising in our East and setting in our West 40 to 60 minutes later each day, an average of 50 minutes as it orbits our counterclockwise-spinning earth, also in a counterclockwise direction. Then for 1 to 2 days each month its brightness at night is 99% + (full moon), and for 1 to 2 days its visibility is less than 1% (New Moon). Click here to see its current timeline in Brisbane.
Click here for more on "sidereal rotation" and apparent retrograde motion with regard to the moon, the planets and the stars.
Click here for the gravitational impact of sun and moon creating tides on the earth.
Click here for a study on the relationship between the moon's phases and rainfall.
Extract from Stephen Wolfram
One thing the Babylonians did was to measure surprisingly accurately the repetition period for the phases of the Moon—the so-called synodic month (or “lunation period”). And they noticed that 235 synodic months was very close to 19 calendar years—so that about every 19 years dates and phases of the Moon repeat their alignment, forming a so-called Metonic cycle (named after Meton of Athens, who described it in 432 BC).
It probably helps that the random constellations in the sky form a good pattern against which to measure the precise position of the Moon. But the Babylonians noticed all sorts of details of the motion of the Moon. They knew about its “anomaly”: its periodic speeding up/getting larger and slowing down/getting smaller in the sky (now known to be a consequence of its slightly elliptical orbit). And they measured the average period of this—the so-called *anomalistic month—to be about 27.55 days. They also noticed that the Moon went above and below the Plane of the Ecliptic the sun's path through the sky (now known to be because of the 5.1° inclination of its orbit)—with an average period (the so-called **draconic month) that they measured as about 27.21 days.
And by 400 BC they’d noticed that every so-called saros of about 18 years 11 days all these different periods essentially line up (223 synodic months, 239 anomalistic months and 242 draconic months)—with the result that the Moon ends up at about the same position relative to the Sun. And this means that if there was an eclipse at one saros, then one can make the prediction that there’s going to be an eclipse at the next saros too.
*Apparently first defined as such in English in 1767
**Apparently perceived as a dragon "eating up" the light whenever the conjunction of the nodal points (the Moon's orbital path twice a month crossing the ecliptic) corresponds with a new moon (resulting in a brief solar eclipse) or a full moon (lunar eclipse).
Click here for a well-written article by Cambridge University on Synodic month (29 days, 12 hours, and 44 minutes), Anomalistic month (27 days, 13 hours, and 19 minutes), Sidereal month (27 days, 7 hours, and 43 minutes), and Draconic month (27 days, 5 hours, and 6 minutes).
**End of notes