It’s often said that Isaac Newton was born on December 25, 1642. That’s true in the fact that Newton’s birth is indicated as that date in historical records, but it is more accurate to note that Newton was born on January 4, 1643. That’s his birthday in our modern Gregorian calendar. At the time of Newton’s birth, England used the older Julian calendar, hence the difference in dates.
The Julian calendar was introduced by Julius Caesar in (what is now) 46 BC. The calendar had 365 days, with an extra day every four years (or leap year). This meant that the average length of a Julian year was 365.25 days. That sounds a lot like the way we do things now, so what makes our modern Gregorian calendar different? The Gregorian year is also 365 days long, with a leap year every four years, but with some exceptions. If a leap year is divisible by 100, (1800, 1900, etc) then there is no extra day. However if the century year is divisible by 400 (1600, 2000, 2400) then there is an extra day. As a result, the average Gregorian year is 365.2425 days long.
This extra complication is necessary because an actual solar year (from vernal equinox to vernal equinox) is 365.2421897 days long. The Julian calendar is a good approximation, but the calendar ran fast by about 3 days every 400 years. Again, not a big deal, but by the time the 1600s rolled around it was about 10 days out of sync with the solar year. The Catholic Church adopted the Gregorian calendar in 1582, and it was quickly adopted by Catholic countries of the day. Protestant countries took longer to adopt the calendar, with England finally adopting it in 1752.
I should point out that when the Julian calendar was adopted, it was known to be slightly off. A century before the Julian calendar was adopted, Hipparchus had calculated the solar year to within 6 minutes of the modern value. But the calendar was easy, and close enough to work well for centuries.