Altmetrics measure the mentions and uses of research. They can complement or provide an alternative to traditional bibliometrics.
In practice, altmetrics look at how many times research is mentioned, used, saved and shared on:
It is worth noting that some altmetrics services also include traditional citation counts in the data they provide.
Altmetrics are becoming increasingly widespread. You are increasingly likely to see them being provided for articles (and other research outputs) you find, and used by academics to help show the value of their work.
As with bibliometrics, they need to be fully understood and their limitations accepted. They are best used in conjunction with other means of evaluation.