Apparently, in the early 1890's, the American Acclimatization Society which was trying to introduce European plants and animals to the US, decided to import hundreds of European starlings into Central Park in New York City. Which they did in fact do, and thus we have the starlings here now, which are listed as a pest species in many states.
Those seem to be the facts. But many folks believe a side story about the starlings is also true. The chairman of the Society at the time, Eugene Schieffelin, thought that all the birds mentioned in Shakespeare's plays should be among those introduced. The European starling is noted in Henry IV, so he made sure they were released in the US.