SInce its origin in the 1960s, measuring a publication's importance or impact has been math-related, counting how many times a publicatin has been cited. This approach developed into measuring a publication's importance by seeing how many times its published articles have been cited relative to the number of articles published, the so-called "impact factor," a service of Thomson Reuters.
With the rise of social media, and its instantaneous and widespread communication, a new approach has arisen, based on how frequently a work has been commented on or referred to in online blogs and other publications, in Twitter, Facebook, Digg, Metafilter, reddit, and other social media services. These are not academic publications, but still provide a measure of importance.
Other alternative measures include the number of times a publication has been viewed or downloaded in from online collections like arXiv or institutional repositories at colleges and universities, the frequency of citation in governmental or policy publications and other "grey literature" publications.
Here are research guides created at other colleges and universities to explain and explore alternative citation metrics, also called altmetrics.