Antonio,
I guess you know that the word "of" implies genetive in English. So, every time you mean "of something", it refers to genetive.
Millions of people = миллионы людей (genitive)
a lot of people = много людей (genitive)
We use genitive even if it's not specified directly:
a few cars = несколько машин/автомобилей (genitive)
We used "единственное число родительного падежа" because a fractional part is always less than one, it's like a part of the entire number (do you see "of"? Ok, use genitive!). So we decline "карат" implying the fractional part of the number, because it's the last part of the sentence.