Metrics that aim to demonstrate the impact of an author over their total research history or within certain time periods can be useful for academic promotion submissions, applying for new roles in academia, and for funding applications.
The h-index is a measure of both productivity and impact of one's published works. It looks at the total number of papers you have published, and the number of citations those papers have received. For a researcher to have an h-index, they must have a certain number of publications (h) that have received at least h citations.
For example, if you've published 2 papers, and each paper has been cited 2 times, you will have a h-index of 2.
What if it's a little more complicated, where your published papers have range of both high and low citations? Let's say you've published 5 papers altogether, one cited 5 times, one 4 times, one 3 times, one 2 times, and one just the 1 time. You will have a h-index of 3.
The limitations: The h-index disadvantages early career researchers as they have not had a chance to publish as many papers as experienced researchers, and it is not appropriate for comparing across disciplines due to varying citation behaviours among researchers.