Learn Hardcore Vietnamese: Chợ này rất đông.

In Vietnamese, demonstratives like này (this) or đó (that) always follow the noun they modify. So chợ này literally is “market this,” but functionally “this market.” English flips the order, so you have to get used to post-noun placement.

What part of speech is rất, and how is it used here?

rất is an adverb of degree, equivalent to English “very.” It intensifies the adjective that follows. In rất đông, rất boosts đông to mean “very crowded.”

What does đông mean in this sentence, and is it an adjective or verb?

Here đông means “crowded” or “with many people.” It’s a stative adjective (sometimes called a stative verb in Vietnamese grammar) describing the state of the market.

Why is there no classifier (measure word) after chợ?

Classifiers in Vietnamese are only used when you specify quantity (e.g. “two tables” → hai cái bàn). In Chợ này rất đông, you’re describing the market, not counting it, so no classifier is needed.

How would you say “This market is extremely crowded” or “This market is too crowded” in Vietnamese?

You have a few options:

Chợ này rất đông → “This market is very crowded.”
Chợ này đông lắm → colloquial “This market is really crowded.”
Chợ này đông quá → “This market is too crowded” (expressing a bit of displeasure).

Can I replace rất đông with rất nhiều người?

Not directly: rất nhiều người means “very many people,” but you need a verb like có (to have/there is) to make a full sentence:
Chợ này có rất nhiều người → “This market has/there are very many people.”
You can’t say Chợ này rất nhiều người without có.

Why is có necessary when I talk about “many people,” but not when I say “đông”?

Đông already carries the meaning “having many people” in one word—it’s a predicate adjective. When you use nhiều người (“many people”), that’s a standalone noun phrase, so you need có to link it: có nhiều người (“there are many people”).

Could I simply drop này and say Chợ rất đông? Would it change the meaning?

You can. Chợ rất đông means “The market is very crowded,” more generic, as if you have one market in mind. Chợ này rất đông specifies “this market” (e.g. the one we’re standing in). The difference is specificity.