In a U.S. election, each state is given a number of 'electoral' votes, based on it population. If a majority in that state vote for a presidential candidate, all of its votes go for that candidate in the election.
A swing state is a state where the presidential support is very close between the candidates and could 'swing' to either candidate.
This is opposed to a 'solid' state that will definitely go for one candidate or the other.