I was watching a TV show and heard them using the expression 凭什么 to say 'why' - I hadn't heard this with this exact meaning before, so I'm wondering what the proper meaning is in English, and what the proper meaning is in Mandarin?
"凭什么” is something similar to "who do you think you are to say i can't do it?" or "you're not qualified to judge me"
凭 is a coverb meaning "based on." If you're looking for a Chinese meaning, my dictionary defines it as 根据 (i.e. based on, according to). In the example you gave, 凭什么 can be understood as "on what basis?", which is only a short jump to "why?"
凭 is commonly used with the word 票, as in:
Admission by ticket only.
Edit: As ziyuang helpfully points out, 凭什么 has a somewhat accusatory tone and implies that the action in question is unwarranted. (That's how "on what basis?" sounds to me in English, but it may not have that sense for everyone.)
"凭什么”has a tone of questioning and a little bit angry,its similar to "by what".For example:"你！凭什么左右我的心情！"means"You! By what can you control my feelings".