Essentially, the h-index is a number intended to represent both the productivity and the impact of a particular scientist or scholar, or a group of scientists or scholars (such as a departmental or research group). This index was proposed by and is named for physicist Jorge Hirsch.
The h-index is calculated by counting the number of publications for which the scientist has been cited by other authors at least that same number of times. For instance, an h-index of 17 means that the scientist has published at least 17 papers that have each been cited at least 17 times. If the scientist's 18th most cited publication was cited only 10 times, the h-index remains at 17. If the scientist's 18th most cited publication was cited 18 or more times, the h-index would rise to 18.
A researcher's h-index can be calculated manually by locating citation counts for all published papers and ranking them numerically by the times cited.
Some databases and tools, like Web of Science, Scopus and Publish or Perish, will calculate this for you.
Calculating the h-index: Web of Science, Scopus, or Google Scholar? A useful guide from MyRI discussing the pros and and cons of using each.
If you apply for an NRF rating, you need to provide your h-index from Web of Science, Scopus and Google Scholar.
Different databases will give different values for the h-index. This is because the databases are selective in their journal coverage and some disciplines are better covered than others. The h-index is based on the citations contained in that database.
Conference proceedings and monographs are not well covered.
Google Scholar (GS) does not provide an h-index but can be used with Publish or Perish software. The data from GS often includes duplicates (pre-prints & post-prints) and it can be difficult to distinguish between authors with the same initials. Manual editing of search results is required.
However, you can set up a Google Scholar Citation Profile which does provide an h-index.
Also, what is considered a good h-index could differ depending on the discipline - a number that is considered low in one area might be high in another field.