For the second time in 21st century, there will be an extra second after UTC 23:59:59 31st December (7:59:59 1st January Malaysia,8:59:59 1st January Japan).If you in Japan, after 8:59:59, there will be 8:59:60.
This is called the leap second.
What is a leap second?
I attached a link with this post (link to wikipedia, again) explaining about leap second.
Easily explained, this leap second is just like the leap year that we know. Just add a day after 28th February.
Adding a second after UTC 23:59:59 is necessary to critical systems like the server for communication, satellite programs, etc etc etc.
For your watch, or clock, you doesn't have to do so. They will run out of batteries soon.