To embroil is to drag someone in to a mess. If you're embroiled, you're in ... DEEP. It's far worse, far messier, and generally far more long-term, than simply being "involved" with something. Nothing good can come of being embroiled.
Embroil can refer to any sort of situation — love affairs, political events, scandals — but it's probably most commonly used in reference to law suits. The classic law suit that embroiled its participants was the fictional one of Jarndyce. v. Jarndyce, in Dickens's novel Bleak House — which went on for so many generations that all the characters' money was eaten up entirely by lawyers' fees. Let us repeat: nothing good comes of getting embroiled.