"More cats" would be «больше кошек».
Больше (with the first syllable accentuated) is a word in the Russian adverbial class, invariable, and means "more." You'll notice that it calls for the genitive case for the noun it operates on.
Большой / большая / большое (with the desinence accentuated) is an adjective that means "big," therefore it agrees in case and gender/number with the noun it operates on. To spice things up a little bit, больше can also be the comparative degree of this adjective: «Моя машина больше твоей.» = "My car is bigger than yours."
Ещё is another adverb that can mean "again," "still," "yet" or "even" depending on context:
- «ещё раз» = "one more time"
- «Он ещё не позвонил.» = "He hasn't called yet."
- «Я его ещё люблю.» = "I still love him."
- «У меня много, но я хочу ещё больше.» = "I have a lot, but I want even more."
- «Дай мне, пожалуйста, ещё чая.» = "Give me some more tea, please." (this is the only context I can think of where ещё alone would be translated as "more" in English.)