They're something you've seen dozens of times, but the reality is that even published authors get compound adjectives wrong from time to time. The following sentence should give you a clue about what I'm referring to:
The derelict red-brick building on the side of the road appeared to be empty. Six rifle-toting guards, however, patrolled the inner corridors in search of intruders. They were beginning an eight-week stint of security, at the end of which the place would be bustling with scientists of all ages and races. The highly trained guards, former military soldiers who had been dishonourably discharged, would eventually become permanent fixtures at the building.
A compound adjective is two words conjoined by a hyphen. By themselves, the two words don't make sense. When joined with a hyphen, however, they act on each other and take on a new meaning. Take, for instance, the first one. If that sentence began, "The derelict red brick building", it would be describing a building that was both painted red and made from brick. That's not what it describes. I wish to convey that the bricks are red, not the building. To do so, I insert a hyphen. This tells my reader that the building is made from red bricks.
"Rifle-toting". The word 'tote' is an informal way of saying that someone is carrying something. On its own -- "rifle toting" -- it reads as though the rifles were carrying the guards. With a hyphen, I'm establishing that all guards are carrying rifles. See how one omission can drastically alter the meaning of a piece? Once you learn that the two words are acting on each other to create, in theory, one complete word, figuring out their usage will become easy.
"Eight-week". Again, another word that acts on the one that preceded it. "I'm going on a two week vacation". So, you're going on two separate vacations, both of which last a week? I don't believe you are. You're going on a 'two-week' vacation. The same applies for days: "I'm taking a five-day leave". Or, "I'm taking a five-month leave". Most people -- in general, employers -- won't be as pedantic as an editor will, so you can get away with this mistake in your letter of request for a paid sabbatical. When it comes to creative writing -- academic, even more so -- it's important to get it right.
"Highly trained". Now we come to the problem that even published authors struggle with. Those two words are acting on each other; one modifies the other. That would dictate that they should be conjoined by a hyphen, wouldn't it? No. 'Highly' is an adverb. Adverbs are words that modify the next word in a sentence. "Stealthily moved", "skillfully created", etcetera. Putting a hyphen in is redundant and unnecessary. The confusion begins with something like this:
The computers had a daily-changing password to prevent being hacked.
Why is this different? Because 'daily' isn't an adverb. It's an adjective. Adjectives follow the above rule. Adverbs don't. That's where the confusion begins. Just consult your dictionary when making compound adjectives. If the word is an adverb, no hyphen.