“Sadness is an emotional pain associated with, or characterized by, feelings of disadvantage, loss, despair, grief, helplessness, disappointment and sorrow.”
I think I can do it more concisely.
Sadness is desire coupled with the understanding that said desire cannot be realized.
Actually there’s one case that doesn’t fit into this neatly, which is empathic sadness. But I think it covers all cases of being sad on one’s own behalf.