Leap years are a consequence of a year not being exactly a whole number of days. Days not being a whole number of hours is a modern innovation, and results in leap seconds being applied certain years since 1972.

There are many different calendars in use throughout the world (Gregorian, Julian, Iranian solar hijri, Islamic lunar hijri, Hebrew...), and historically there have been many more. These calendars have different rules for dealing with leap years, for example:

• The Julian and Gregorian calendars add an extra day to February based on a fixed calculation.
• The Iranian calendar adds an extra day based on observation of the vernal equinox.
• The Jewish calendar is lunisolar. Most of its months have fixed length, but two of them have variable length. This depends on a fixed calculation that keeps the months synchronized with the moon, and keeps certain holidays from being on inappropriate days of the week (on / next to Shabbat). There are 12 or 13 months each year, also based on calculation.
• The Islamic calendar is lunar, but based on observations of the moon rather than calculations. The months don't take place at the same time every year because an Islamic calendar year is shorter than an astronomical year.

Most of the world now uses the Gregorian calendar, which has leap years on years which are divisible by 400, or by 4 and not by 100. It was based on the Julian calendar, which has a leap year every 4 years. The goal of the Gregorian calendar was to closely match observation of the solar year, and specifically to calculate a more historically-accurate date for the Easter holiday by correcting the drift of the vernal equinox since the Council of Nicaea in CE 325. To do this, it dropped 10 days. Interestingly, this was based on observation and not by calculating the calendar so that it would match in CE 325, which would have given only 9 days (in 500, 600, 700, 900, 1000, 1100, 1300, 1400, and 1500).

Therefore the Gregorian calendar immediately differed from the Julian calendar, and adopting it changed the reckoning of the current date. The first areas to adopt were the Papal states; Spain (and thus Portugal and much of Italy); Poland and Lithuania. As decreed by Pope Gregory (followed by King Philip II and Queen Anna Jagiellon), in those countries the change occurred at the end of Thursday, 4 October 1582 (Julian), being followed by Friday, 15 October 1582 (Gregorian).

The difference between the Gregorian and Julian calendar can cause historical confusion, mainly because it was not adopted by every country at the same time. The most well-known example is that Russia did not adopt the Gregorian calendar until 1918, with January 31 followed by February 14. As a result, the October Revolution (which started on October 25, 1917, Julian) was in November as reckoned in the West. The Russian Orthodox church, and most other Eastern Orthodox branches, still use the Julian calendar to calculate the dates of Christmas and Easter. So Coptic and Orthodox Christians celebrate these holidays on different days from Catholics and Protestants.

Some countries also had years starting on days other than January 1, and changed it to January 1 either when adopting the Gregorian calendar or at other times. For example, until 1752, the English, Welsh and Irish used the Julian calendar except that the first day of the year was March 25. In 1752 it was changed to begin on January 1, making 1751 only 282 days long, and the Gregorian calendar was adopted later in 1752, making it 355 days long (11 days slip, but 1752 was a leap year). The Scottish had used a Julian calendar beginning January 1 since 1600, and also adopted the Gregorian calendar in 1752. This change also affected British overseas colonies, including the American colonies.

The Julian calendar itself replaced an earlier Roman lunisolar calendar, which added a 13th month every 2-3 years. This intercalary month was theoretically added based on observation, but it was apparently done in a somewhat haphazard manner by the council of priests (pontifices) and was apparently lagging. As a result, Julius added two additional intercalary months to 46 BCE (Roman year 708) for a total of 15 months (and 445 days!). As far as I'm aware, there is no canonical explanation for the exact number of months or days he added.

As for whether the current date is inaccurate: this obviously depends on what you hold as your standard of accuracy. But the answer is probably that it's not perfectly accurate. The Gregorian calendar does not perfectly match the length of the year (365.2425 vs the true length of ~365.24219 days). So dates will drift very slightly, but only by ~3 hours since the calendar's introduction. More importantly, its exception-based leap year rule is a looser approximation than adding a leap year every 4-5 years, so the equinoxes and solstices don't all fall on the same day.

There is also the matter of year numbering. The West uses the Common Era (CE) which is a secularization of the Christian "Anno Domini" ("Year of our Lord"), supposedly dated with 1 being the year that Jesus Christ was born. This was calculated by Dionysius Exiguus in 525, and its accuracy is a whole nother can of worms (to sum up: probably not accurate). It gradually became standard in the West from about 731 CE (Bede's Ecclesiastical History of the English People) to 1422 CE (when Portugal adopted it).