Figure Analysis
Vashti doesn't get a speaking part in Esther, but her actions speak much louder than her words. King Ahasuerus wants her to come see him so that he can show her off to all the subjects in his kingdom, but she delivers a crushing diss, refusing to show up. Obviously Ahasuerus can't stand this supposed insubordination—so he fires her from her queenly job, which Esther attains later on.
We don't really know why Vashti refused her husband's request, the same way we don't know why people do a lot of the things they do in Esther. Still, it's easy to guess at some of the possible reasons. Ahasuerus is being inappropriate; he wants to show his queen off to his subjects in a way that is potentially demeaning, treating her as a sex object. The first chapter of Esther says that he's been partying seven days and is "merry with wine" and he wants his eunuchs to bring Vashti so he can "show the peoples and the officials her beauty; for she was fair to behold" (Esther 1:10-11). So, that's one very plausible answer.
Post Queen Career?
Ahasuerus divorces Vashti and orders her to never come before him again—which he likely thinks is a punishment, but probably actually is a relief for his now-deposed queen. After this, Vashti disappears from the narrative. We have no idea what a fired queen gets to do afterward: whether she's stuck in prison, or gets an extended vacation in exile, or just has to hang out and find a normal job.
Vashti is replaced by Esther who takes a different approach to the king. Namely, she puts up with him, humbling herself before him in order to try to ultimately sway him into thinking in her direction. It's sort of a subtler tactic than the earlier queen's refusal to humor the king—but Vashti, nevertheless, remains a quite admirable figure in the course of this story.