Why do we have leap years?

A year is the time Earth takes to orbit the sun, and that is not a whole number of days. It is approximately 365.2422 days. A calendar of exactly 365 days therefore drifts by nearly a quarter of a day annually, which accumulates to a full day every four years and a month every century or so. Left uncorrected, the seasons would gradually shift relative to the calendar.

Adding a day every four years corrects most of it, but overcorrects slightly, because the true figure is a little less than a quarter. The Julian calendar used exactly this rule and drifted by roughly three days every four hundred years, which by the sixteenth century had accumulated to about ten days.

The Gregorian calendar refined the rule. A year is a leap year if divisible by four, except that century years are not, unless they are divisible by four hundred. So 1900 was not a leap year and 2000 was, which surprises people who only know the first part of the rule and caused a genuine class of software bugs.

That refinement brings the average calendar year to 365.2425 days, within about half a minute of the true figure, which will not accumulate to a full day for several thousand years.

The extra day sits in February for historical reasons rooted in the Roman calendar rather than any astronomical logic.

A separate and unrelated adjustment is the leap second, occasionally added to account for irregularities in Earth's rotation rather than its orbit. That is a different problem with a different solution, and the practice is being phased out.