An idiom meaning to be unable to act on equal terms with someone — e.g., to be no match for someone, or to feel indebted/obliged to them and thus deferential.
長年お世話になった先生には頭が上がらない。
I can't hold my head up against the teacher who's helped me for years (I feel indebted).
あの先輩は仕事ができて、頭が上がらない存在だ。
That senior is so competent that I can't compete with them.
彼にお金を借りているから、頭が上がらない。
Because I borrowed money from him, I feel indebted and can't act on equal terms.