To embroil is to drag someone into a mess. If you're embroiled, you're in deep. Being embroiled is 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 — romantic entanglements, political events, scandals — but it's probably most commonly used in reference to lawsuits. The classic lawsuit that embroiled its participants was the fictional case 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.