This question is close to several Frequently Asked Questions. In particular, your question about birthdays has already been asked many times. I know only a few of the basics of daily timekeeping (why we have 24 hours, how ancient people kept track of time), not enough to write up a proper answer. However, I can answer as to how the calendar year was conceived of.

This varied a lot from culture to culture, and from time period to time period. Tropical Rainforests are really the only climate on Earth that doesn't experience much seasonal variation. For societies outside of these though, seasons would have greatly influenced crucial elements of life: Particularly the temperature, the weather, and the availability of food.

The common four-season calendar isn't arbitrary. Summer days are longer and hotter while Winter days are shorter and cooler. In areas with a large amount of seasonal variation in climate, keeping track of which season it was, and how long it was until the next season, was critical to survival. The oldest known calendar was made in Scotland circa 8000 BC.

This calendar-monument, as well as similar calendar-monuments built along the Nile later on, was aligned with the Winter Solstice (December 21st), which correlates with the shortest day of the year. It is consistent, occurring at the exact same time every year. Early calendar systems were almost always based in part on the phases of the moon, which also follows a regular cycle of about 27.32 days.

The early Roman Calendar, the ancient ancestor of our modern calendar, was quite similar. The months were based on the phases of the moon, but as these do not correspond with the length of the year, additional days were added based on astronomical events (particularly the solstices) to match the calendar with the true length of the year.

Both the Roman and Egyptian calendars were later modified to be based purely on annual events. The Egyptian calendar came to be based on the heliacal rising of Sirius, and the Roman calendar came to be based on the Spring Equinox. as explained well in this thread by /u/aescolanus, the Roman transition from the Spring Equinox to January 1st was a result of the awkward conversion from the convoluted Roman Lunisolar calendar subject to intense yearly variation to the regular Roman Solar calendar.

The length of the year does not perfectly correspond to a whole number of days, and this issue was causing both the Roman and Egyptian calendars to fall out of sync with the seasons. The Egyptians noticed that their calendar rapidly stopped correlating exactly with the heliacal rising of Sirius, leading to the concept of the "Sothic Cycle", a 1461 calendar-year period until the Rising of Sirius once again corresponded with the start of the year. Roman writer Censorinus notes in De Die Natali, written 238 AD, " The Egyptians the space of four years is shorter by one day than the space four natural years, and a complete synchronism is only established at the end of 1461 years."

The transition from lunisolar to solar was huge for both the Egyptian and Roman systems. The next major change occurred with Roman Dictator Julius Caesar's "Julian Reform" in 46 BC, which added a leap day once every 4 years. This change was also applied to the Egyptian Calendar in 26 BC by Caesar's nephew and successor, Augustus.

The Roman calendar would not be further modified until the Gregorian Calendar Reform in 1582 AD, which removed leap days from certain years they would otherwise apply in order to fix the far smaller but nonetheless significant gap between the length of the Julian Year and the astronomical year. Even the modern Gregorian Calendar does not perfectly correspond, but it will be a millennium before it falls out of sync by even a single day.