Pope Gregory XIII adopted a calendar which kept track of time based on the year of Jesus Christ's birth. This went along swimmingly for many, many years, especially as other systems of measuring time were ignored by the Christians.
Then, in this century, people built computers to think for them1. These computers were instructed that there were no years before 1900 or after 1999. So, come the end of 1999 (shortly after Jesus's birthday) the computers will start over, telling us that it's the year 1900.
So as we approach the end of the year 1999, people are rushing round trying to fix the problem.
Nobody is quite certain how big a problem the Millennium Bug will actually be. Some people are stockpiling toilet paper while others are blowing their noses with total abandon. A sustained theory is that the only machines that will truly be affected by the Y2K bug are time machines. This would explain why nobody has visited us from the future to allay our concerns.