Adding a fixed number of days to a date is simple counting: move forward day by day, rolling over into the next month or year whenever you cross a month or year boundary. Adding months or years is a fundamentally different kind of operation, though, because months don't have a fixed length, and treating "add one month" as "add 30 days" produces the wrong date more often than it produces the right one.
Why adding a month isn't the same as adding 30 days
The correct way to add a month to a date is to keep the day-of-month the same and increment the month (rolling into the next year if needed), not to add a fixed number of days. March 15 plus one month is April 15, not April 14 or April 16, regardless of whether March has 31 days and April has 30. If you instead added exactly 30 days to March 15, you'd land on April 14, a full day off from the intuitively correct answer, and the error compounds if you keep adding 30-day increments across months of different lengths.
The month-end edge case
The trickiest part of adding months is what happens when the day-of-month doesn't exist in the target month. Adding one month to January 31 can't produce "February 31", since February never has 31 days, so the calculation has to fall back to the last valid day of the target month instead, February 28 in a common year or February 29 in a leap year. This is exactly the kind of situation where the leap year rule from our guide on why leap years exist directly affects the result of an otherwise unrelated calculation. Different systems resolve this month-end overflow slightly differently (some clamp to the last valid day, others roll forward into the next month), so it's worth checking which convention a given calculator uses if you're relying on the exact date.
Adding years is usually simpler, except for one date
Adding or subtracting years typically just changes the year while keeping month and day fixed, since every year has the same twelve months. The one exception is February 29: adding a year to a leap-year February 29 lands on a date that doesn't exist in a non-leap year, which is the same underlying issue anniversary calculations for leap-day birthdays have to solve, covered in more detail in our guide on anniversary and birthday countdown math. For quickly verifying any of these calculations, or working the problem in the opposite direction (given two dates, how many days apart are they), the Add/Subtract Dates Calculator and Date Difference Calculator cover both directions.