The sequence of dates of the UTC second markers will be:
2005 Dec 31, 23h 59m 59s 2005 Dec 31, 23h 59m 60s 2006 Jan 01, 00h 00m 00s
The difference between UTC and the International Atomic Time TAI is:
from 1999 Jan 1, 0h UTC, to 2006 Jan 1 0h UTC : UTC-TAI = - 32s from 2006 Jan 1, 0h UTC, until further notice : UTC-TAI = - 33s
In other words, the year will be longer by one second just before midnight, new year's eve, GMT. That's 4:00 PM PST, 7:00 PM EST.
Please take that second to just go nuts. One second. All to yourself. Use it wisely.
I also recommend reading Paladin of the Lost Hour by Harlan Ellison.
The reason leap seconds are needed is because there are two different definitions of the length of a second: the solar time that you use every day, where the there are 60×60×24 seconds per day (meaning the length of the second varies as the speed of rotation of the earth varies) and the engineering definition, where a second is defined as 9,192,631,770 oscillations of a cesium-133 atom: the atomic clock.
Mostly humans are concerned with time as it relates to the comings and goings of the Sun, but you need to be a little more accurate if you expect that satellite to be where you left it, so you need the length of your unit of measure to not change. That would affect everything else, too, since the official definitions of other units like the volt and the meter are based on the second. (You thought the meter was still defined by the length of that platinum-iridum bar in France, didn't you? Get with it!)
So the time system we use day to day, UTC ("Coordinated Universal Time") is a compromise between solar time (UT1, "Universal Time", formerly GMT, "Greenwich Mean Time") and atomic time (TAI, "International Atomic Time"): in UTC, seconds are of a fixed length, but leap seconds are periodically introduced to prevent "solar" noon and "atomic" noon from drifting away from each other.
But really it's all a lot more complicated than that!
And then of course there was the switch from the Julian to the Gregorian calendar at various dates (mostly) between 1582 and 1752, in which 4-Oct-1582 was immediately followed by 15-Oct-1582, in order to correct for the inaccurate leap-day-insertion algorithm in the Julian system, which over the centuries since its introduction had caused the calendar to detach from the observed equinoxes by ten days.
It can be really hard to know what time it is.