To put it as briefly and simply as possible: Purim, which means "lots" (as in lottery) in Hebrew, tells the story of cousins (or uncle and niece, depending on what you believe) Mordecai and Esther, two Jews who were living in a Persian kingdom in the 5th century BC.
The king, Ahasuerus, put out a call for a new wife after his current spouse, Queen Vashti, refused to show off her beauty for the people — there are a few interpretations of these events — and ended up choosing Esther, who didn't disclose her religion.
Separately, Mordecai discovered a plan to kill the king and saved his life (though the king was not told). Mordecai, sitting outside the palace gates, refused to bow to the king's viceroy, Haman. Haman was so offended by this that he decided to kill not just Mordecai, but all the Jews living in the kingdom. The king allowed this because he didn't know about Esther's heritage or Mordecai's heroism. Haman picked the date of the extermination by ... wait for it ... using lots, and the date was set for the 14th of Adar.
Mordecai implored his cousin/niece to ask the king for mercy, and Esther eventually obliged after praying and fasting. Esther, during a feast, revealed her Jewish heritage to the king and asked that her people be spared. The king, furious at Haman, allowed Haman to be hung from the gallows he had constructed for Mordecai, and the Jews were saved from extinction yet again, l'chaim.