What do I do if a teacher fails me after they said I'd pass? I just failed a university-level course that the instructor assured me I was passing.  We've discussed my grade in emails but we continue to disagree on how it should be calculated and even on what assignments should count. (The instructor did not adhere to the grading policy in her syllabus - this is a passing/fail issue.)

I have proof that I did enough to pass.  Grades are in and the school is out for the holiday break, but I feel like I need to do something before this becomes permanent.  What should I do?  As a student, how do I challenge a grade I think is unfair if can't resolve it with my instructor?
Contact your teacher (email) and start a paper trail where you outline the assignments in question: when they were due, when they were submitted, and if the teacher received them.
Given the break, schedule an appointment with the teacher after the break to sit down and discuss the issues at hand.
(As per aeismail's comment) I would agree to escalate to the department chair and inform your academic advisor before the dean.
At the same time as 5, should you choose to and if your institution has such a department, contact your ombudsperson office to create an informal avenue of conflict resolution outside of formal (and time consuming) solutions. They offer a means of conflict resolution that tries to arbitrate issues before you take formal action.
Depending on the results (no meet, meet but no success) escalate by appealing to your college Dean the issues and ensure that all paperwork is in order: an executive summary documenting the chronological order of events, all of your correspondence with the teacher indicating that you abided by the syllabus.
If and when you have a hearing, dress nice and present your case when you are called in for an appointment with the dean. Be factual and punctual. Don't demean or insinuate personal characteristics of the teacher in question, stick to the facts and what is the issue at hand.
At the end of the day, if all else fails, I would stand to believe that your college Dean answers to a higher 'officer' of the school. There should be a means to escalate your appeal to this final authority. There might be more, but for most if not all issues, this would be the highest authority to appeal to.