The reason skype stopped working is that Microsoft decided to change its communication protocol in an incompatible way, starting from 1. August.
In order to continue working, all devices running skype (computers, smartphones, etc.) on all operating systems must update to the latest version of skype for their platform. For Linux, its version is 188.8.131.52.
As far as CentOS 6 is concerned, you want to download the latest "dynamic" skype tarball from the skype website , and install it as per instructions on the CentOS wiki . Essentially, the installation is the same as for previous versions, no surprises.