Altmetrics, as the name suggests, are alternative metrics to the traditional impact factor and other indices. Altmetrics may be disseminated from a wide range of social media networks, such as blogs, Twitter, and Mendeley.
The official definition from Altmetrics.org:
"altmetrics is the creation and study of new metrics based on the Social Web for analyzing, and informing scholarship."
In terms of showing evidence of scholarly impact, using altmetrics in addition to the traditional metrics can offer a comprehensive view of one's work. The Altmetrics.org site has more information about altmetrics, how they work, and how they can benefit your scholarly impact analysis.
These sites include altmetric tools for determining your impact across cyberspace.
Use these to obtain data by journal, such as impact factor and h-index.
Use these for basic counts of the number of times an article has been cited.
Use these tools to identify article citation counts as well as other data, such as downloads, discussions, etc.
Use these tools to generate data by author for activity such as promotion & tenure review and candidate searches.