Definition:

A compound adjective ....

I an adjective that contains two or more words joined by a hyphen (-)

I saw a man-eating alligator.

Examples:

I saw a man eating alligator.

We are describing the alligator. What type of alligator is it? It is one that eats men (or people).

I saw a man-eating alligator.

This sentence without the hyphen sounds like a man is eating an alligator.
(man is the subject, eating is the verb, alligator is the object or thing that is being eaten).

I saw a man eating alligator.

More examples:

I live in an English-Speaking country

I live in an English-Speaking country

She is a world-famous singer

More examples:

She has a big blue book.

No hyphen between two adjectives