Science Citation Index or SCI is a Reverse Index for references in published Scientific work. It answers the question
Who has cited this key paper from the past?
It used to be in huge tomes of small print and is now available electronically
at www.isinet.com 
. -- [Broken link]
Summary: I found the link above to be just an annoying tease. -- John Dowd
Details: While the link above tells all about the Science Citation Index, and had many links to related citation indices, I could not find anywhere that I could actually USE the thing to search for anything. It just seemed to endlessly go on about how great and powerful it all was, with so many journals, etc.
But there's no there there. No shopping cart either, just a link for how to contact them for purchasing a product. So--all very nice to be sure, and the thing itself is indeed great--I have used the paper version in research libraries to great effect long ago. But ultimately not useful.
Now that it's electronic, it does more than just answer the question above; it's a fully operational database of the topology of the scientific literature. As well as finding which papers cite paper X, you can find which papers it cites (handy if you don't have a copy yourself), and do clever things like looking for the core papers in any field by looking for papers which are cited by many of the papers cited by some particular paper in the field you're interested in (or something).
The science citation index is the primary database used for Citation Analysis, the voodoo by which scientific papers and journals are rated.
There are other citation indexes besides the Science Citation Index. Cite Seer, in particular, is a free, online citation index for computer science which even includes full text of most of the papers it covers.
See original on c2.com