Y2K is a computing problem caused by the way computers store dates. In these cases, the year 2000 would be represented as 00 rather than 2000, which causes a problem whenever a calculation is based on that date.
More examples are available.
For more information see the SAS and ISC Year 2000 web pages. A failure in January 2000 could interrupt your research efforts, your teaching efforts, or administrative efforts in support of your research and teaching. Such problems may be trivial, but if dozens or hundreds of little glitches happen at the same instant, it could take weeks or months to find staff time to assist you in remediation. NSF and NIH awardees are generally responsible for ensuring Year 2000 Compliance for their projects. This responsibility extends to anticipating and reacting to events such as the Year 2000 and taking all steps necessary to mitigate potential problems that might be caused by the Year 2000.