Ibn Khaldun (1332-1406) was observing and trying to explain the rise and fall of dynasties in the part of the world he was most intimately familiar with, namely Islamdom.

The influence of his best known work, the Muqadimmah or Prolegomena, is perhaps two-fold. One is that Asabiyyah seems to be a pretty plausible explanation for the political dynamics he was observing, so he continues to be cited in descriptions of dynastic turnover in the region.

The second, and as I understand it more recent phenomenon is the wider re-evaluation of Ibn Khaldun's work in other areas of political science, sociology and philosophy. I'm not sure I can comment on "how influential" he has been; I'm not familiar enough with the political science tradition to comment on how widely read he has been by, say, Weber or Marx or whomever.

But there has been a very wide recognition of the level of sophisticated political thinking he demonstrated which anticipated a huge number of aspects of later political thinkers. That being said as far as I'm aware those later thinkers are not believed to have been directly influenced by him per se.

If you're interested there's a great In Our Time podcast on Ibn Khladun, http://www.bbc.co.uk/programmes/b00qckbw

And the Muqaddimah itself is a great read.

edit: deleted a bit about a particular translation as, though it's the one I'm familiar with, is not well regarded.