I need to know what down in this specific sentence means. I don't know if it is a preposition or an adverb.
You pays your money and you takes your choice:
Traditional grammar calls it an adverb: a word which modifies words which are not nouns.
—But down plays an obligatory role in this sentence; I see no sense in which it can be said to "modify" put.
Some contemporary grammarians call it a particle:
a word that
does not belong to one of the main classes of words
is invariable in form, and
typically has grammatical or pragmatic meaning.
—SIL Glossary of Linguistic Terms
—In other words, particle is a box where we dump anything which doesn't belong anywhere else. (At one time adverb served this function, but that use is no longer chic.)
Other modernists call it an intransitive preposition—essentially a word which can serve as a preposition phrase all by itself, without an object.
—I like this. Down in your sentence behaves just like any other preposition phrase (on the table, in your pockets, behind your ears) would, as a complement to the verb put depicting the goal where the object of the verb ends up.
Put your pencils down.
The example that you give makes use of a multi-word verb, a phenomenon that is especially common in the English language. According to the Cambridge dictionary, "down" in your sentence would be an adverb particle, the best of both worlds of StoneyB's response. In fact, "down" is on the list of the most common adverb particles.
Specifically, "put down" is a phrasal verb, a classification of multi-word verbs. Phrasal verbs commonly take objects, and in the given example, the object would be
or more strictly speaking, the noun
modified by the possessive adjective "your."
The Cambridge Dictionary gives a very similar example:
Take your shoes off.
where the direct object
splits the main verb
and the adverb particle
of the phrasal verb "take off."
Check out this link for more information on multi-word verbs: Cambridge Dictionary.