OpenSSL is a software package and developer library that programmers have used to create the systems used for encrypted communications on the majority of servers. These communication channels include a variety of services and protocols, the most common is HTTPS which is typically enabled when you purchase an SSL certificate from someone like Comodo, VeriSign/Symantec or GeoTrust.
When you access a website via HTTPS (for example https://chrisburgess.com.au), and see a padlock in your browser address bar, this indicates that a server is using a protocol called SSL (or technically SSL/TLS). A large portion of all servers using SSL rely on OpenSSL for this functionality. The OpenSSL vulnerability has nothing to do with whom you purchased the certificate from, but rather the underlying server software.
This post only covers a very simple overview of this problem, for more information please visit Heartbleed.com.
How serious is Heartbleed?
Very. In very simple terms, this vulnerability potentially allows visitors to your website to access server memory, including data that should not be accessible.