The question of who invented the first calendar is very hard to answer -- no one knows for sure. Many ancient societies had them. The Egyptians are, however, credited with the first solar calendar (based on the sun as opposed to on the cycles of the moon).
The reason we have a leap year is much easier... As you know, a year is defined as the time it takes the earth to complete one orbit of the sun. Sadly, the earth doesn't take a whole number of days to orbit but instead takes 365.25 days to orbit the sun.
So, every four years we add an extra day to make up for the fact that our year is actually one-quarter day shorter than it should be.
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