Refers to one's social or moral obligation, or the honor and standing one must uphold. Often used in contexts of fulfilling one's role or maintaining face.
He did his utmost to fulfill his duty.
For samurai, preserving their honor was more important than anything else.
one field; a branch; a division
The exact derivation is uncertain. The kanji 一 (one) and 分 (part, share) together suggest 'one's allotted part' or 'one's proper share,' which came to mean one's duty or honor.