Two calendars, and a jump
The Julian calendar adds a leap day every fourth year, so its average year is 365¼ days, slightly longer than the year of the seasons. Over the centuries the dates slipped, and in 1582 Pope Gregory XIII's reform dropped ten days and changed the rule: century years are leap years only when they divide by 400, so 1700, 1800 and 1900 were not, but 2000 was.
Countries changed at different times. Britain and its colonies waited until 1752, when Wednesday 2 September was followed by Thursday 14 September. The weekdays carried on as normal; only the dates jumped. Russia changed in 1918 and Greece in 1923. The Julian–Gregorian converter has the full table.
Old Style years and dates like 1720/21
Until 1752 the legal year in England began on 25 March, so a date between 1 January and 24 March carried the previous year's number. Writers often gave both: 1 February 1720/21 is 1 February 1721 as we count today. This finder reads every year as starting on 1 January and tells you when a date falls in that window; the converter has a box for years copied straight from an Old Style document.
What the Julian Day Number is
Astronomers number days in one unbroken count from noon on 1 January 4713 BCE (in the Julian calendar), ignoring months and years. Each day has one number, whichever calendar a writer used, which makes it the safest way to compare dates. Every conversion on this site goes through it.