When you count a number of items, you say "Okay, zero apples, one apple, two apples, three apples..."  Right?

  Of course you don't.  You start at one.  You start counting at one.

  Well, what makes you think the years are any different?  They most certainly are not different.  They are the same.  Years start at one.

  It was long ago decided, that, instead of counting what year it was based on changing events (like the selection of a new king or emperor), the years would be marked by the reign of our Lord Jesus Christ.  So it was decided that we would call the years before "B.C." (Before Christ) and the years after "A.D." (Anno Domini, or Year of the Lord, from Latin) There is also the heading B.C.E. (Before Common Era), but nobody much uses that one in normal conversation.

  When the A.D. / B.C. thing was decided, early in the first millennium, the years before Christ were counted backwards, and ended at the year 1 B.C.  After that, the year 1 A.D. started.  There is no year 0.

  The first 100 years, or the 1st century, started at year 1 and ended at year 100.  That's a common feature of all centuries - they start at '01' (1, 101, 201, and so forth ) and end at '00' (100, 200, 300, and so forth).  So the 1st century was year 1 (A.D.) to year 100 (A.D.)  The 2nd century started at 101 and ended at year 200.  The 3rd century started at year 201 and ended at year 300.   The 4th century started at 301 and ended at 400...

  Now, if you'll notice, the century's name has a direct correlation to the LAST year of the century.  The 3rd century ends in the first year labeled 300.   The 5th century ends in 500.  The 16th century ends in 1600.

  You'll also notice that the bulk of a century does not have the number of that century in it.  The 1st century (1-100) doesn't start with a 1 (1x100) (100) until its last year.  The 14th century doesn't start with 14 until its last year; it goes from 1301 to 1400.  It may seem odd to us that the 14th century goes from 1301 to 1400 (for example, to me it sounds odd to say "1394 is in the 14th century")  But the fact is, a century starts at something 1 (1, 101, 201, 1301) and ends at something 0 (100, 200, 300, 1400).

  Now, when does the 20th century end?  Why of course, the last year of the 20th century is not 1999.  The last year is 2000.  Remember, the last year of the century is the first time a year starts with that number.  The last year of the 5th century is the first time a year starts with 5 -- the year 500.  The last year of the 17th century is 1700.  The last year of a century is never "whatever" 99 -- it is always "whatever" 00.  If a century began at 0, it would end at 99.  But they don't start at 0, because there is no year 0 A.D. or B.C. or anything.  No year 0.  By the same token then the 50th century, say,  will go from 4901 to 5000.  (End year; 50 x 10²)

   When the 19th century (1801-1900) ended, and the 20th century began  (in 1901) there was a big discussion of this same subject I'm relaying.  At that time, most of the world (at least the part on the Western world's calendar) proclaimed that 1901 was the first year of the 20th century.  As they should have.  (And if you're following my line of reasoning, you already know what I'm leading up to.)
 

  Well, when is the 20th century?  It is, indeed, the year 1901 to the year 2000.  The 20th century ends in 2000.

  So, when is 1000 years?  Of course, as with a century, it starts at 1 and ends at 0.  The first thousand years, the 1st millennium, went from 1 to 1000.  That means the 1st millennium ended the first year  starting in x * 1000.  The year 1000.  The 2nd millennium goes from 1001 to 2000.  Millennia start at 1 (1, 1001, 2001, 3001...) and end at 0 (1000, 2000, 3000, 4000...)

  We are now preparing for a millennial celebration.  The only problem is that the last year of the 2nd millennium is....

  Right.  The last year of the 2nd millennium is the year 2000.  Just as the 2nd century ended in 200, the 2nd millennium ends in 2000.

  So, when does the 3rd millennium start?  If you've been paying attention...

  Yes, the 3rd millennium begins in 2001.
 

    So, if you celebrated in 2000, you celebrated the end of the last millennium, not the start of this one!