The H-index is the number of articles with at least h citations. Kim McDonald writes that the H-index for an author describes "a scholar with an index of h has published h papers each of which has been cited in other articles at least h times."
Several databases calculate the H-index for authors based on the citations they know about:
You can also calculate your own H-index. View the chart below for a quick visual.
There are several indices related to the H-index; there are good descriptions of these on Harzing's Publish or Perish site.
Egghe's G-index was designed to improve upon the H-index by adding weight to higher-cited papers.
The Individual H-index gives less weight to coauthored papers. Harzing describes 3 variations on the Individual H-index.
The Contemporary H-index gives less weight to papers cited in earlier years, so you will see a lower score come from calculating this metric on the body of your research over time.
The purpose of the M-index (or M-quotient) is to measure productivity over time. To retrieve this metric, divide the H-index by the number of years you have been publishing research.
The i10-Index is the number of publications with at least 10 citations. It can be found in Google Scholar in the My Citations area.