"There are nine tables."
Numbers (and kanji in general) have different readings in Japanese. The character for nine (九) can be read as きゅう or く, and these are called the On'yomi, the readings borrowed from Chinese pronounciation, but it can also be read ここのつ, which is the Kun'yomi, the original Japanese pronounciation. If you use a number to count nouns, you need to use particles called “counters”. In this case we have the つ particle, which is the particle for objects, and here the Kun'yomi reading is needed.
"arimasu" and "imasu" are both verb that can be roughly translated as "exist"
the difference between the two is that "armiasu" is used for inanimated objects and "imasu" for animate ones.
Ex: "inu imasu" - there's a dog "tsukue arimasu" - there's a desk
sorry for the lack of kana.
あります means "exists" and can also qualify existence, as in location or count. です is called a copula, a word used to equate two concepts, usually two nouns or a noun and an adjective. The topic of です can be left out and inferred from context, such as "it" or "I." Both あります and です translate to "is," which can be used for either purpose in English.
"です" comes after an adjective like "かわいいです" (it is cute)。あります and います are verbs meaning "to exist". And have the ます ending which follows verbs.
I'm sure there is a better way to explain this, I'm still learning the language too. I just remember that from high-school Japanese class. :p