A compound modifier is a single phrase – two words, sometimes three – that enlivens the noun it precedes: long-bearded hermit, six-headed grotesquerie, cardamom-laced crème brûlée, top-tier media.
Compound modifiers require a hyphen because without it the words fail to act together as a single descriptive and then, well, all hell breaks loose. Okay, maybe not, but things do get awkward and confusing.
When Aaron, in "Titus Andronicus," says, “What, what, ye sanguine, shallow-hearted boys!” he’s not referring to shallow boys or hearted boys, but shallow-hearted boys. When F. Scott Fitzgerald said “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function,” he wasn’t talking about a first intelligence or a rate intelligence but a first-rate intelligence. Without the hyphen the meaning is lost and the noun that follows is less interesting and colorful.
What’s worse, sometimes the meaning of the entire sentence is messed up. “He was a man-eating hippo!” tells me that the hippo eats people, “He was a man eating hippo!” tells me that a man was eating hippo. Big difference.
And yes, this being English, there are exceptions to the rule: