How it is calculated
A researcher has an h-index of h if h of their papers have each been cited at least h times. Someone with an h-index of 12 has twelve papers with twelve or more citations each. It was proposed by Jorge Hirsch in 2005 as a single measure combining productivity and impact.
Its design is deliberate. It resists both a large body of uncited work — publishing a hundred papers nobody cites does not raise it — and one exceptional paper, since a single highly cited work can only ever contribute 1.
That second property is the one people find counterintuitive. A researcher whose single landmark paper reshaped a field can have a modest h-index, and the metric is behaving exactly as designed.
One extraordinary paper adds 1 to your h-index, the same as a paper cited exactly h times. The metric is built to ignore outliers.
What it cannot tell you
It cannot be compared across fields. Citation behaviour differs by an order of magnitude between disciplines — typical counts in pure mathematics and in molecular biology are not on the same scale. An h-index quoted without a field is uninterpretable.
It cannot be compared across career stages. The h-index can only rise, and it rises with time in the field. Comparing a researcher five years post-PhD with one twenty-five years in measures career length as much as anything else.
It says nothing about contribution. A middle author on many well-cited collaborations and the person who led the work can end up with similar figures. And it counts self-citations and citations from close collaborators the same as independent ones.
Using it sensibly
State the source and the date whenever you quote it. Google Scholar, Scopus and Web of Science will each give you a different h-index for the same person because they index different literature — Scholar counts the most sources and generally returns the highest figure.
Give field context alongside it. A percentile against comparable researchers in your subject area communicates far more than a bare number, and is a normal way to present bibliometric evidence.
And use it with other things rather than instead of them. Independent citation counts, evidence that others adopted your methods, invitations that reflect standing in the field — these say things the h-index is structurally incapable of saying.