@stanmann is correct. The numbers 2, 3, and 4 take the genitive singular. The numbers 5-20 take the genitive plural.
I'm still confused though because "a blue cup" is just "one cup" and it's not in the nominative singular form. I, personally, would use in the singular nominative in Russian, but I trust that the makers of this course know better. I just don't understand why it's not in the aforementioned form, but I do understand why "two plates" are in the genitive singular, though.
Maybe because синюю чашку is the accusative, because there is an implied "I want" when you are ordering things.
I'm not certain, but I seem to remember that a noun following the number 2, 3, or 4, when the number is in either the nominative or the inanimate accusative, takes the genitive singular. But I also realize, that this old fart's memory is not what it used to be. So don't take this as gospel.