means that something is preferable.
It comes from 寧[むし]ろ, literally meaning 'preferable'
Often in combination with より
むしろ、安い方を買って方が良いと思う。 Preferably (rather than that), buying a cheaper one is better.
俺はMMOよりむしろシングルプレイヤーのゲームの方が好きだ。 (rather than) MMOs, I prefer singleplayer games
何だそっちが怒ってるの？むしろこっちが怒りたいよ。 Why are you mad? Rather than you, I should be the one (who wants to be mad).