Or, more accurately, about as much as happens to us when one of our centuries ends - people worry about banking software crashing (aside: the ancient Mayans did not have banking software), and then nothing happens.
David Stuart, a Maya researcher, explains it on his blog: Baktuns and more Baktuns:
This upcoming 188.8.131.52.0 date is a repetition of the “base” of the system which fell in 3114 BCE, also represented as 184.108.40.206.0. Back then, the subsequent bak’tun number was re-set as 1 (220.127.116.11.0) and thereafter their count progressed forward until the reappearance of 13 bak’tuns on December 21 of this year. This repetition of 13s has led some to suppose that a similar re-set of the bak’tun system is upon us now, and that we are destined to go back to 18.104.22.168.0 in some 400 years from now. This is not true. Based on texts from Palenque that project calendar stations far into the future, we know there will be a linear sequence of bak’tuns from here on, represented as 22.214.171.124.0, 126.96.36.199.0, and so on. This will run forward still until 188.8.131.52.0, about 2,400 years from now.
Notice that at the end of this roughly 13,000-year span that the bak’tun changes to 0 and the next higher period, the piktun, turns over as 1. As it happens, the piktun unit before this date was set at 13, although this is left unwritten in the dates above. (Mayanists have long tended to just write five numbers of the Long Count, following the convention of the ancient Maya scribes themselves. But we know that this is a truncated representation, and that there were many more cycles above bak’tun and piktun. The full system I call the “Grand Long Count” encompassed 24 units!)
So, yeah, nothing. From this NPR article, Stuart also said:
"It's a big deal — if you're an ancient Maya astronomer priest," Stuart says. "But apart from that, they didn't say anything about ... what will be happening."