Learn Russian: Я заказал дюжину чашек для кафе.

Дюжина is a regular feminine noun meaning a dozen. Because it is the direct object of the verb заказал, it takes the accusative case ending -у, becoming дюжину. Like the other numerical nouns in this lesson, it requires the items being counted to be in the genitive plural, which is why we see чашек.

Why does the genitive plural of чашка have an е in the middle?

To form the genitive plural of most feminine nouns ending in -а, you drop that final vowel. Dropping the -а from чашка leaves an awkward consonant cluster -шк at the end of the word. To make it easier to pronounce, Russian inserts a fleeting fill vowel, usually о or е. Here, it becomes чашек.

Could I just use the number двенадцать instead of дюжину?

Yes, you could say двенадцать чашек. However, using дюжина is very natural and common when ordering standard sets of items like cups, plates, or eggs, exactly like in English. Grammatically, двенадцать is a numeral and would keep its nominative-looking form for an inanimate accusative object, whereas дюжина acts like a standard feminine noun.

Why does кафе not have a genitive case ending after для?