Discord is the strife and tension that arises when two sides disagree on something, like the high-pitched screaming of two kids fighting over the front seat of the car.
Discord can be broken down into the prefix dis, meaning "different," and cord, which stems from an old word for "heart." So that leaves us with "different hearts." So if we're talking about music, discord means a lack of harmony — tones that clash so badly your ears bleed. But when there's discord between people, their hearts are in different places — which usually results in more than a few raised voices clashing disharmoniously.