Many computer systems - some even dating from the first commercial machines and some very modern ones - still use only 2 digits to record the year in any date records. Thus, January 1st, 1984 is recorded as 01/01/84, or 840101, or some similar 6 character format.
So far, working in the 20th century, each system merely assumes that the century is 19. What happens on the first of January 2000? Well, for example,an age calculation may show a negative value; or, rent calculation might show 99 years and 11 months credit. Other possibilities exist - accounting systems refusing to issue cheques to pay bills, or, refusing to accept entries because they are not for the current year - 1900.
Accounting systems are especially interesting. By law, account details must be maintained for 5 or 7 years for tax purposes. These will have dates with 6 character fields. Your updated accounting system - with 8 character date fields - will not be able to read them... You'll need to maintain two systems until you can legally (and comfortably) discard the old data.
The following pages highlight some of the problems, point to other sites detailing problems and finally point to some ways of overcoming the problems.