The TL;DR version: There is a tiny gap between traditional "rotational time" and the modern "coordinated universal time," due to the fact that the earth spins a tiny bit slower every year due to tides and internal processes. No clock can accomodate an "extra second," so clocks are stopped for one second at 23:59:59 in order to add the "leap second" to our universal time.
Elisa Felicitas Arias of the International Bureau of Weights and Measures explained in 2008 that "It might seem stupid to say that you have a difference of only one second... but for the stock exchange, one second is important. For an airport, one second is important. For global navigation satellite systems, the difference of a second is unacceptable."
On his blog about working around potential leap second disasters, Marco Marongiu explained that even though the leap second has been added many times since it was introduced in the 1970s, many systems and applications have still not quite adjusted to the change. Here's another post, this one from Google, explaining how they account for the change to insure their products and services remain unaffected.
However, many of the most popular sites and applications on the internet were not prepared for the leap second adjustment, and many of them saw widespread outages as soon as the adjustment took place.
Reddit, Gawker media sites, StumbleUpon, Yelp, FourSquare, LinkedIn, and Meetup were among the many sites affected for some time.