"There are seven tables."
「あります」is a verb, the polite form of the verb 「ある」which is an existence verb for inanimate objects, often translatable as the English expression "there is".
「ます」 is the ending that is used in the polite form of almost all verbs, an exception being the verb "to be", which has the polite form 「です」.
I personally found this easier to understand when learning casual speech first and then moving to polite forms, which is the way most native Japanese speakers learn (people usually speak to kids in all casual speech and young kids usually speak back in this way, from what I have been told by native speakers).
For some reason most modern courses teach the polite forms first, which seems a little unnatural to me. It's like, yes, it's good to be polite by default, but it's not the simpler linguistic form and not as clear for teaching the language.
です is a copula indicating distal style, and though it is usually translated to English as "to be", it is wrong to consider it as a verb with a meaning "to be". It is not a verb at all and doesn't have a meaning besides its function of forming distal style nominal predicates. It's easy to see from how negative distal style nominal predicates are formed: you add a negation of the real "to be" ある.
Negative adds the negation of the real verb "to be":
では above is a "gerund of direct style copula だ"+は and is not translated.
I answer 七つのテーブルがあります and I think this should be right (i even get accepted in other question with the same structure).