Calendar Dates in English
The question "What day is it today?" might generate the answer "the second".
To make matters worse, people can't even agree when each week starts. In the United States most calendars start the week on Sunday, not Monday.
We use "in" when we talk about weeks.
Example: In the first week of September I'm flying to Paris.
The weeks of our calendars are based on the cycles of the moon. It takes about 29.5 days to pass through all of the Moon phases. In Mesopotamia people observed a 28-day cycle and divided that into four separate 7-day parts. They also used leap days to correct for the discrepancy between the actual number of days (29.5) and the observed number of days (28).
Humans could see that the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn weren't fixed in one single position in the sky. They "moved". And these became objects of worship. The story of days, weeks, months, and calendars is unequivocally linked to religion.
And that's why there are two different ways to organize your week. Starting the week with Sunday is based in religion, whereas starting the week with Monday is not.
We use "on" when we talk about days.
Example: I have to go to work on Monday.
Example: We are leaving on the 3rd of August.
Example: I saw him on the first day of the week.
Sunday was named for the Sun. In most religions the creation of the universe and of humans is associated with the Sun or light. The ancient Egyptians had a "day of the sun", which they passed on to the Romans who passed it on the Germans who passed it on to the English. In China, Russia, and many other countries Monday is the first day of the week and the word for Sunday is not named for the Sun.
Monday is named for the Moon in English. Tuesday is named for the Norse god of war Tiwaz. Wednesday is named for the Norse god Odin. Thursday is named for the Norse god Thor. Friday is named for the Norse goddess of knowledge Frigg. And Saturday is named for Saturn.
We use "in" when we talk about months.
Example: School starts in September.
Years are strange too. Once every four years we have an extra day: February 29th. We call the year that this happens a leap year.
A typical year according to our calendars today is called a "common year" and lasts 365 days. But that is not a complete orbit around the Sun. When the Earth makes a complete orbit around the Sun it's called a "tropical year" and it actually lasts 365.242189 days (365 days, 5 hours, 48 minutes, and 45 seconds). But in our modern calendar system, the Gregorian calendar system, once in four years we have a "leap year", which lasts 366 days.
If we didn't add a leap day on February 29 every four years, then each year would begin about 6 hours too early (before the Earth has completed its orbit). Leap days fix this error by making sure that our calendar aligns with the Earth's orbit around the Sun. Roman and Chinese calendars have also included leap months for the same reason.
We use "at" when we talk about time.
Example: The party starts at 10 o'clock.
Example: At 9AM I'm going to the doctor's office.