One of my favorite genres of tv and movies is crime drama. There are constantly characters demanding other characters to either “put the gun down” or “put down the gun”. I’m confused as to which usage of “down” is correct. Can you please explain which usage is correct?

Answer 1

both are correct and commonly used.
In both cases, "down" is an adverb.

It can be confusing, though, since we say
walk up the stairs
where "up" is a preposition.
No one says walk the stairs up.

On the other hand, we do say
walk the dog up (that is, up the stairs)
but not: walk up the dog (only a flea might do that)

But, wake up the dog
as well as wake the dog up.

We get some of our grammar from German, which has what are called separable prefixes. When actually used, the prefix is usually moved to the end of the clause. That's why it often looks like we are ending the sentence with a preposition, while it is often just an adverb.

One of my favorite sentences in this regard is what the child said when his father came upstairs to read him a bedtime story, but it was the wrong one. The boy said, "What did you bring that book I didn't want to be read to out of up for?"

See whether you can parse that!

Answer 2

You're dealing with phrasal verbs -- that is, commonly used verb phrases that include a verb and a preposition or adverb (or two) and have meanings of their own.

Here's a good explanation and some very good charts. Some phrasal verbs are separable ("put down" in your example) and some are not -- and sometimes a phrasal verb contains 3 words! Be sure to scroll down and check out all three lists on this webpage:
http://guidetogrammar.org/grammar/phrasals.htm