The Collaborative International Dictionary
millennium bug \mil*len"ni*um bug`\ (m[i^]l*l[e^]n"n[i^]*[u^]m b[u^]g`), n. (Computers) An error in the coding of certain computer programs which store the year component of the date as two digits, assuming that the first two digits are 19, rather than as a complete number of four digits; when such programs are used after January 1, 2000, the date may be misinterpreted, causing serious errors or total failure of the program; -- called also year 2000 bug, year 2000 problem and Y2K bug.
Note: In the several years leading up to the year 2000, large corporations and other users of computers in total spent many billions of dollars correcting this error in the programs they use.
Wiktionary
n. (context computing English) A design flaw in computer systems which represented the year with 2 digits (ex. 1984 = 84), which could cause date calculations to fail after 1999.