When the northern pole of the world points towards the sun, it is the warm seasons of the year in the northern hemisphere, and when it points away from the sun, it is the cold season; but today, and on the day of the Autumnal Equinox, the Earth's axis is in-between. The northern pole of the world is now turning back to the sun, lengthening the days in this hemisphere, and indeed this penitential season is called ‘Lent’ in English, from the Anglo-Saxon word lœnan, meaning ‘to lengthen’.
Fittingly, since this is the beginning of the season of growth, we find that many ancient calendars set the Vernal Equinox as the New Year's Day. But the observation of the equinoxes goes back to remote antiquity, and I am always amazed at the astronomical accuracy often found among the ancients throughout the world.
The date of the Spring Equinox is nowadays somewhere between March 19th and the 21st, but traditionally it was observed on March 25th; this change was due to a poorly-implemented calendar reform, where leap-year days were mistakenly inserted for a while every three years instead of every four years. Julius Caesar was assassinated the year following his calendar reform, but Caesar Augustus corrected this error after 36 years, and dropped several leap year days.
Since religion is concerned with higher things such as first and final causes, while government tends to be concerned with opinions and appearances, the calendar traditionally is the domain of the priesthood. Julius Caesar reformed the Roman calendar by his authority as pontifex maximus, or head of the Roman religion, and his calendar is still in use by Orthodox Churches. Likewise the Gregorian Calendar we now use was promulgated by Pope Gregory XIII.
Designing a calendar is rather problematic. It is useful and fitting that a calendar synchronizes with the various natural cycles, such as the orbit of the Earth around the sun, the seasons, the lunar phases, and day and night. Calendars often incorporate weekly cycles used for market days, days of rest, and religious observances, including our seven-day week. The problem stems from the fact that these cycles are not numerically commensurate, in that a fixed number of months, weeks, and days cannot be precisely fit into a solar year or lunar month, and so a calendar has to be a compromise.
The Christian calendars synchronize with the seasons and do not observe the lunar cycle - the ancient Roman calendar originally had each month start with the New Moon, but that was quickly changed to fixed months. The Jewish calendar synchronizes with both the Moon and the seasons, but this requires an additional month inserted into the calendar periodically. The Islamic calendar is lunar but does not synchronize with the seasons. All three include the seven-day week, but the week does not have any particular association with months and years.
The cycle of the seasons do not match the orbit of the Earth around the sun, due to the phenomenon of the Procession of the Equinoxes. The ancient Egyptian calendar would start the New Year based on the rise of Sirius, the Dog star, and this would drift with time.
The Christian dating of Easter is related to the Vernal Equinox. The day of the Resurrection of our Lord occurred on the Sunday following Passover, and so ecclesiastical calculations of the day of Easter roughly approximates the Jewish calendar. However, early churchmen rather wanted the observance of the Resurrection to be held on the same day throughout the world instead of strictly following the Jewish calendar, which often varied by community, was otherwise somewhat unpredictable for future years, and was not under the authority of the Church. Logically, Easter is the first Sunday after the first Full Moon on or after the Vernal Equinox, which is in the Jewish month of Nisan; but our calculations are only approximate.
The English word ‘Easter’ comes from the pagan deity Ēostre; this was first observed by Saint Bede the Venerable, and the pagan custom had already died out at the time of his writing. However, the name of this Christian feast day in most other languages is derived from the Hebrew Pesach (פֶּסַח), or Passover, which is where we get the English word Paschaltide.
The Catholic love for places, seasons, and things often lead to accusations of paganism, but I think this attitude is perhaps due to our contemporary split between rationalism and spirituality. On the one hand, we have a scientific worldview of planetary orbits divorced from God, and on the other we have iconoclastic religion devoid of art and forgetful of history. Rather, the Church appreciates all of Creation, which was made very good, although it is fallen. Echos of this attitude may perhaps be found in the lyrics of Loreena McKennitt's lovely song The Mummers' Dance, where “the springtime of the year” is “the work of Our Lord's hand”.
See also my post on the numbering of days in the Roman calendar.