Learn Turkish: Yeni garson iki kahve fincanı getiriyor. - The new waiter is bringing two coffee cups.

In Turkish, when you put two nouns together to make a compound noun, the second noun needs a possessive suffix. Kahve (coffee) and fincan (cup) combine to make kahve fincanı (coffee cup). The -ı is this possessive suffix, matching the vowels in fincan.

Could I just say "iki kahve fincan" without the extra suffix?

No, because kahve is a noun, not an adjective. If you used an adjective like yeni (new), you wouldn't need a suffix on the noun: yeni fincan (new cup). But because you are combining two nouns (coffee + cup), you must use the compound structure and add the suffix to the second word: kahve fincanı.

Since there are two cups ("iki"), why isn't the word plural like "fincanları"?

In Turkish, a noun that comes after a number always stays singular. Because of this rule, you say iki kahve fincanı (literally "two coffee cup"), never iki kahve fincanları.

The cups are being brought, so they are the object of the sentence. Why isn't there an accusative case ending on "fincanı"?

The sentence mentions "two coffee cups," which is an indefinite amount (just any two cups). In Turkish, only specific direct objects (like "the cups" or "those cups") get the accusative case ending. Since these cups are indefinite, they stay in their plain compound form. If he were bringing "the" two specific coffee cups, it would be iki kahve fincanını.