A Swing state is a State in the United States of America that can vote either Democrat or Republican depending on how popular the parties are there. States that regularly vote Republican are called Red States, the Bible Belt states are examples of Red States. States that regularly vote Democrat are called Blue States, the cool New England states are examples of Blue States. Because swing states are neither clearly red nor clearly blue they are sometimes called Purple states. Swing states are also called Battleground states.
The election system used in the United States, the Electoral College, determines the winner not by popular vote, but by counting in which states the candidate has a majority, with a bonus given to the small states. As a result, the states where a certain candidate is known to win are pretty much ignored by the candidates -- it doesn't matter if the candidate wins 51% or 99% of votes, they still get all the votes from the state. This makes the swing states matter much more; in fact, winning the whole election depends on the candidates winning in the swing states. Not only that, but the system can result in the candidate winning the election while losing the popular vote -- which was exactly what happened in the 2000 Presidential Election.