What is a Research h-index?

Typical h-index plot

An example researcher would have an index of h if he has published h papers, each of which has been independently cited least h times. The bottom line here is that the h-index combines the number of publications and the number of citations per publication into a single metric.

The major caveat here is that it only works properly only for comparing academics in the same field since citation density differs widely among different fields.