Here was the thinking: Since most software was written to store and/or manipulate dates using only two digits for the year, calculations built upon these dates might not execute properly because the computer will not see dates in the 21st century as being larger numbers than those in the 20th century.
(For example: 2000 - 1998 = 2, but 00 - 98 = -98 or 98 if the application does not allow negative numbers.)
The result of this might have been that your accounting software would see all accounts receivable as overdue, due to the fact that no customers had paid in 98 years!
The two-digit date convention assumed that the century is always "19." This assumption was regarded as a necessity in the early days of commercial computing because of the high cost of computer storage and memory.
The leap year issue seemed to be a bona fide concern as well, leap years are calculated by a simple set of rules, but there were computer systems and applications that did not recognize the year 2000 as a leap year so many people thought this would cause all dates following February 29, 2000 to be offset incorrectly by one day.
The rules for leap year calculations are as follows: a year is a leap year if it is divisible by four, but if it is divisible by 100 it is NOT a leap year, but if it is divisible by 400 it IS a leap year. Thus, the year 2000 was a special case leap year that happens only once every 400 years!
The third main Y2K problem was found in older code bases. In order to write more efficient code which allowed for the use of less memory, date fields were sometimes used to provide special functionality. The most common date used for this was 9/9/99. In some applications the use of the special date meant "save this data item forever" or "remove this data item automatically after 30 days," or "sort this data item to the top of the report." Within each organization, special date codes may have been used differently, and this illustrates one of the main reasons why no single tool could locate all uses and/or misuses of date data.
And there you have it! Fortunately nothing dramatically wrong occurred, instead the whole earth seemed to be celebrating that night ;-)