An author's H-Index is calculated by counting the number of publications for which an author has been cited by others at least that same number of times.
The purpose of the H-Index is to remove focus from the uncited or overly-cited works that might give a skewed picture of a author's impact.
Source: "The H-Index: Calculation, Pros and Cons" (Linkedin.com)
Google Scholar, Scopus and Web of Science are the most used databases for calculating H-Index. Links to these databases and a list of their limitations can be accessed here.