Refers to a lack of harmony, whether in music (dissonance) or in relationships/opinions (discord). Often used in formal or abstract contexts.
You can feel dissonance in that chord.
Discord has arisen between their opinions.
Compound of 不 (fu, 'not') + 協和 (kyōwa, 'harmony, concord'). 協和 itself is from 協 (kyō, 'cooperate') + 和 (wa, 'peace, harmony').