When Unix(tm) time was being created, a line was drawn in the sand from which all time was to be measured from. This is called the Epoch. Unix(tm) time is measured in seconds since this point in time: January 1, 1970.
Epoch is centred around +0000, GMT, and hence this click over will be at the same point in time for everyone, unlike New Year's eve which follows the sun around the planet.
More information can been gleened from this Slashdot article or this document on critical unix times.
The Epoch will roll over on January 19, 2038.