Counting days is simple. The day/night cycle can be observed everywhere on Earth (except sometimes at the poles).

Counting months is more complicated. Though initially based on the lunar cycle of 29.5 days, one could not round off to 30 because 12 months of 30 days (360 days) do not add up to a year.

All of the difficulties arise from a year that does not contain a whole number of days. The Earth turns around the Sun in 365.2425 days, or practically 365 + ¼ days (365.25). If one ignores this ¼ day, one accumulates discrepancies which can end up placing the month of January in the middle of the summer (after 800 years).

It was in 46 BC, under the reign of Julis Caesar, that the leap year reform was undertaken. For this reason, this calendar  was named the "Julian" calendar.  It was again reformed in 1582 by Pope Gregory XIII.

The Earth is tilted on its orbit. This causes an unequal amount of sunshine to fall in different parts of the planet during the course of a year, and this is responsible for the seasons. You can use this animation  to illustrate the  variations in the length of daylight,  and the seasons,  in the northern hemisphere.

The circles represented are:

1. Arctic circle (66°N)
2. 45°N latitude (Seattle, Toronto, Milan)
3. Equator
4. Antarctic circle (66°S)

The animation is not drawn to scale for either sizes or distances. Nor are the relative speeds of rotation and revolution accurately represented.

The motions of the Moon around the Earth and of the Earth around the Sun are complex. The motions involved in revolutions are superimposed on the movements involved in rotations. The Earth and the Moon both turn on their own axis (rotation), but both also move around another object (revolution).

The rotation of the Earth (24 hours) explains the alternation of day and night.
The revolution of the Earth around the Sun (365.25 days), because of the inclination of its axis (not shown here), explains the changing seasons.

The fact that the rotation of the Moon on its own axis and its revolution around the Earth both require the same amount of time (29.5 days) explains why the Moon always shows its same side to the Earth.

In this animation, neither proportions nor time scales are accurately presented.

The motions of the Moon around the Earth and of the Earth around the Sun are complex. The motions involved in revolutions are superimposed on the movements involved in rotations. The Earth and the Moon both turn on their own axis (rotation), but both also move around another object (revolution).

The rotation of the Earth (24 hours) explains the alternation of day and night.
The revolution of the Earth around the Sun (365.25 days), because of the inclination of its axis (not shown here), explains the changing seasons.

The fact that the rotation of the Moon on its own axis and its revolution around the Earth both require the same amount of time (29.5 days) explains why the Moon always shows its same side to the Earth.

In this animation, neither proportions nor time scales are accurately presented.

The plane of the Moon's orbit is tilted at about 5° with respect to the ecliptic (the plane containing the Earth's orbit and the sun).
This animation illustrates why a waxing or waning crescent moon seems to be tilted at a much larger angle in the sky.

Never stare at the sun without proper eye protection.

An eclipse of the Sun can only occur at New Moon when the Moon passes between Earth and Sun at the correct distance. (The Moon’s orbit around the Earth is slightly elliptical).
The corona can only be safely viewed without the solar filter during the few brief minutes of totality --  that is,   when the Moon blocks the entire face of the sun.
The rest of the time, you have to use the appropriate solar filter and never look directly at the Sun without proper eye protection.

Relative orbits for the planets of the solar system are shown. Their proportions are respected (except for the Sun) but not the scale ratio ! Pluto's orbit is elliptical and out of the ecliptic plane.

Mars's trajectory seems to periodically go backward. This is called a retrogradation (or retrograde motion). Because the Earth orbits 1.88 times faster than Mars, the relative motion of both explains this phenomenon. Here we are observing in the geocentric frame of reference. See also "Retrograde motion #1" to have the heliocentric point of view, which is the correct one with which to study this phenomenon.