負けず嫌い describes someone who cannot stand to come off worse — in a game, an argument, an exam, or just a comparison. It is a standing character trait, not a mood, so it is normally predicated of a person or a 性格 rather than of a single occasion.
Formation: 負けず 'not losing' (a literary negative of 負ける) + 嫌い 'dislike' — literally 'hating not to win'. The apparent double negative is a historical quirk; the meaning is simply 'hates losing'.
As a na-adjective it takes な before a noun:
The word is usually approving or at worst affectionate — it suggests drive and persistence, and is a stock compliment for athletes and hard-working students. To criticise the same trait a speaker would reach for something else, such as 'a sore loser'.
Compare 勝ち気, a near synonym focused on the wish to win rather than the dislike of losing, and used more often of women in older writing.