Feminine nouns ending in -а or -я lose that ending in the genitive plural. Frequently there is an extra -e- or -o- shoved inbetween the last consonants to facilitate pronunciation. The ending for the genitive plural is called a "zero-morpheme" (like: кошк-ø, which then surfaces as кошек). The genitive plural of книга, for instance, is книг.
I recall learning that to say you have no cats, it is sufficient to say "У меня нет кошки" because the genitive singular effectively is saying "I have none of cat." So if you are asked "У тебя есть кошка?" you could respond "Нет, у меня нет кошки." In what context would one say "У меня нет кошек." ? Would it be if someone asked about a quantity greater than one, such as "У тебя много кошки?" ?