A scientist has index h if h of his/her Np papers have at least h citations each, and the other (Np - h) papers have no more than h citations each.The h-index discounts researchers who have one or two highly cited articles or books, or those researchers who just churn out mediocre papers.
There are loads of problems with the h-index. Google scholar and other citations counters are inaccurate because of trouble parsing and disambiguating papers. Citation counts do not accurately measure the quality of the paper—a paper that opens a field will get many more citations than a paper that closes it. The h-index rewards fads and cliques who always cite each others work. The h-index gives greater weights to more senior scientists and doesn't separate those who had good early careers from those still going strong.
Having said that we do love to compare ourselves with our colleagues in any way possible. An automated calculator does not work well for even mildly common names but it works great for "Fortnow" and while my h-index of 23 does not put me among the h-number elite, I'll take it.