Describes a person or situation that is easily upset, offended, or requires careful handling. Often used with な to modify nouns.
He has a sensitive personality, so be careful with your words.
This issue is delicate, so let's handle it carefully.
The kanji 機 (opportunity, mechanism) and 徴 (sign, indication) combine to suggest a subtle, easily triggered nature. The exact derivation is uncertain; the word is rare and may be a coined compound.