A swing state is an American state in which both candidates in a Presidential election have a reasonable chance of winning, because support is almost evenly divided. The opposite of a swing state is a “safe state,” in which one candidate has a demonstrably clear majority, ensuring that the state will go to him or her. As one might imagine, swing states tend to receive a great deal of attention during election years, because they can make or break the outcome of an election.
There are a number of reasons why the citizens of a swing state have such divided loyalties. Often, one candidate has a stronghold in a particular region of the state; classically, a Democratic candidate is supported by urban residents, while a Republican candidate can capture a large segment of the rural vote. Because these populations are often roughly equal, the state ends up being divided. In other instances, the distribution may be much more even, with most counties split roughly down the middle in terms of who they support.
You may also hear a swing state called a purple state, in a reference to the red and blue colors used on maps to identify election results; the implication is that the swing state is so perfectly split that the colors blend to turn the state purple, instead of decisively red or blue. Swing states are also known as battleground states, in a reference to the immense amount of money, times, and resources sunk into such states in the hopes of winning the vote; the margin in a battleground state can sometimes be very small, so it's an instance where every vote counts.
As a general rule, swing states are identified very early in the election cycle, allowing candidates to focus specifically on them, often largely ignoring the safe states. Politicians might make a token visit to states which support them early on in the election cycle, but they invest the bulk of their campaign time in the swing states, mobilizing volunteers and staffers to get out the vote in their favor while holding rallies, attending events, and generally putting themselves in the public eye.
Because swing states are so important, they are often a topic of interest in the larger community as well. The results of opinion polls and primaries in these states are followed closely in the news, in the hopes of garnering information about how these states might go in the general election. People also travel from all over the country to work in swing states in the months and weeks leading up to the election, supporting their candidates with phone calls, door to door canvassing, and other techniques.