Ahasuerus is translated as Xerxes in more modern versions of the Bible such as the NIV. Ahasuerus and Xerxes I are the same person. For more about Xerxes, see here.
Vashti is a queen of the Persian Empire, married to King Xerxes I (called Ahasuerus in the Bible.) Her story is what sets the stage for the book of Esther. During a drunken party she is called out to dance for the king. When she refuses, the king, with some coercion from his guests, decides that it is time to find another queen.
It is speculated that Vashti is not just fighting for women’s rights in her refusal of the king. It is thought that she might have been asked to dance naked for the party guests to show off her beauty. She refused to be humiliated in this way, which is the real force behind going against the king’s wishes.
Vashti ends up being banished from the king’s presence and a search for a new queen was launched. Esther would end up becoming queen at Vashti’s loss. Although one might expect that Vashti was executed for her refusal of the king, this doesn’t happen.