Databases that provide citation counts include:
The H-index is a measure of the number of publications published by an individual and how often they are cited
An author's h-index, or Hirsch index, is the number (integer) n for which the author has published at least n papers which have each been cited at least n times. Eg
How to find your h-index
Scopus - Select author search or ORCID. click view Citation Report
Google Scholar - Create a Google Scholar profile which will generate your h-index
Publish or Perish - Based on Google Scholar Metrics this free software can calculate a variety of author metrics
Microsoft Academic - re-activated in 2016, Microsoft's answer to Google Scholar
Impact factors attempt to measure the importance of a journal based on how often its articles are cited in the field.
Journal Impact Factor (JIF)
SNIP (Source Normalized Impact Per Paper)
SciMago Journal and Country Rank (SJR)
Limitations
Obvious factors can heavily influence journal impact factors, such as journal title changes, or publishers gaming the system by requiring authors to cite articles from other journals by the same publisher. However there are more systematic problems:
What are altmetrics? (from Altmetric.com)
Symplectic Elements - as well as standard metrics such as SNIP and SJR, Elements provides Altmetrics e.g. number of times an output has been picked up by news outlets, tweets, blogs, Mendeley, CiteULike, Facebook etc
Ebsco databases (e.g PsycInfo, Business Source Complete) now include PlumAnalytics
Some journal web-sites also provide altmetric statistics