If your h-index score is 241, that means you have at least 241 papers/articles that have been cited by at least 241 other papers/articles.
Any database could measure an h-index from its citation data.
Currently, Web of Science and Scopus (look in Author Details) display it.
Dr. J. E. Hirsch created the h-index in the paper "An index to quantify an individual’s scientific research output" in 2005.
He asked and answered "How does one quantify the cumulative impact and relevance of an individual’s scientific research output?"
A scientist has index h if h of his or her Np papers have at least h citations each and the other (Np h) papers have h citations each. In this way the h-index is intended to take into account both the number of research papers an author as produced and how many times their papers have been cited.
See also
Ball, P. 2005, "Index aims for fair ranking of scientists.", Nature, vol. 436, no. 7053, pp. 900.
An example of the use of h-index is the list of top researchers in computer science, ranked by h-index score.
Data that is used for citation measurements include: