When you're embroiled in something, you're tangled or mired in it — in other words, you're involved so completely that it's hard to get away from it. You might be embroiled in an argument between two friends, unable to step back and be neutral.
Two warring countries might be embroiled in peace talks, or two characters on your favorite TV show could be embroiled in a scandal. In either case, people are tangled in a complicated situation or a serious conflict. The earliest meaning of embroil was "throw into disorder," from the French root embrouillier, "entangle or confuse."