This is a long long story, going back to the founding of Rome in the 700s BC.
The very first Roman calendar (itself borrowed from the Greeks) had ten months, starting with Martius and ending with December:
Martius
Aprilis
Maius
Junius
Quinctilis
Sextilis
September
October
November
December
These months covered only 304 days of the year. The remaining 61 days, after December, were not part of any month.
Martius was the time of the Spring equinox, and Spring was a common time for ancient societies to start their year. (However, it's not clear why the Roman calendar started about 21 days before the equinox.)
So, the original Roman calendar, from which our modern Gregorian calendar is descended, did start in Spring.
However, this 10-month year didn't last very long. Within about 50 years, two extra months were added to the year, to make it up to 355 days: Januarius and Februarius (the 10 missing days per year were made up every second year by a "leap month" of 20 days, known to the Romans as an "intercalary" month).
The Romans now had two beginnings to the year: Januarius was the month that each year's consuls started their year of service, but the calender year was considered to start in Martius. (This is sort of like the modern practice of having a financial year start on a different date to the calendar year.)
It wasn't until Julius Caesar's time, nearly 700 years later, that the New Year was moved to the first day of Januarius (Caesar did this as part of his reform of the calendar, which became known as the Julian calendar).
The Julian calendar continued to be used in most of Europe long after the Western Roman Empire fell, but with no consistency about when the start of the year was - it started on different dates in different countries: 1st March; 1st January; 25th March; even 25th December. (Even now, the English tax year starting on 6th April reflects their previous New Year date of 25th March, adjusted for the Gregorian calendar.)
Eventually, the confusion pushed everyone to consistency. This was assisted by the gradual adoption across Europe of the Gregorian calendar, with its New Year start date of 1st January (going back to Julius Caesar's earlier decision).
The current choice of January 1st is therefore based on Julius Caesar's earlier choice of this date, which was based on the fact that the month of Januarius "contained the festival of the god of gates (later the god of all beginnings)".
From here. You may also be interested in the 'The year and months' section of the Popular Questions pages (as found in the sidebar).
I've found the same question asked only 2 hours earlier.
EDIT: I fixed the link for the 'The year and months' Popular Questions section.