TA/student
Issues with grading - The best thing you can do is prevent grading conflicts in the first place. If the grading of assignments is at your discression, give the students a clear rubric for how you are grading. This helps you be consistent and prevents students from accusing you of unfair grading. If there are weekly assignments, it's a good idea to photocopy all assignments for two reasons: (1) if you ever lose the original, you always have a back-up and (2) if the student brings back their assignment for a regrade that looks suspicious, you can see if they added things post-grading. It's always a fine line between standing firm and being receptive to a student's concerns. Some students are especially fussy - they'll argue for half an hour over 1/2 a point on a 100-point test. Remind them that in the grand scheme of things, 1/2 a point is really not that important.
Harassment - Try to be nice and win your students over. This will make the entire semester so much nicer for you, and if a student decides to verbally abuse you, you will have the rest of the class on your side. Often times, they are the best answer to one student's misbehavior. However, if someone is being really inappropriate, addressing the problem promptly and professionally is usually the best course. Remember, if the student disrupts class, they are preventing other students from getting knowledge for which they've paid dearly. It's not fair to you and it's not fair to your students. If a student is harassing you outside of class, get your lab manager or professor involved so they know what's going on. Professors are usually very supportive and can offer the best course of action.
In addition, you can find more details at Interacting with Students
TA/TA
One TA explains a course concept that conflicts with your explanation - Talk to the other TA before you tell the student that the other TA is wrong. You may be able to clear up misunderstanding. If you're wrong, your students will not think less of you for saying so (no, really), and the other TA should enjoy the same confidence.
Another TA grades differently than you - Usually each section is normalized, so students with a TA that grades generously do not have an advantage over students with a more rigorous TA. Again, a clear rubric of how you will grade can prevent a lot of conflict.
Student/professor
Professor makes a mistake in grading or lecturing - Talk to the professor first (depending on your rapport, it may be better to do so in private). Usually, they will happily correct themselves. If not, gently point out to your students some "alternatives."
Student complains about professor’s tests, teaching style, etc. - You may want to gently pass on these comments to the professor, though they may not be well-received. You may also suggest that you forward your students' complaints via anonymous emails, so the professor can get more direct feedback. The professor should be aware that students are ultimately the ones writing the evaluations.
TA/professor
Professor offers regrades that skirts TA’s grading rubric - This is perhaps one of the most frustrating circumstances. You may try and talk to the professor, and tell him/her that their regrading undermines your authority and the consistency of the students' scores.
Professor makes poorly-written tests and grading keys - If you can, ask the professor for clarification. Unfortunately, the professor may not be available for questions when you grade exams. In this case, divide points in a clear rubric so you can be as consistent as possible.
