"there are no cats here"
"there are no small dogs left."
negate a singular noun phrase
"there is not a cat here"... we usually use the contraction "there isn't a cat here"
watch out for the context though...
"there isn't any [small dogs] left." any is singular here.
not is mostly an adverb
"he is not running five miles"
"they are not running five miles"
"i am not running five miles"
it can also negate whole sentences,
"not all men dance" vs. "all men dance"
"the not so handsome guy"
"it's not under the car"
"it's not mine"
"it's not his"
"it's not joe's"
Additionally, "not" is an adverb, so it pairs up with a verb, especially auxiliary verbs, like forms of BE (is, am, are, will, was were) and modal auxiliaries (can, should, would, might, must). Contraction is a good test:
Verb + "not"
EX: He is not running five miles. / He isn't
EX: It is not under the car. / It isn't
EX: It is not mine. / It isn't mine.
EX: It is not Joe's. / It isn't Joe's.
EX: They are not running five miles. / They aren't
EX: It can not be true. / It can't
EX: She does not like fish. / She doesn't like fish.
EX: They should not leave. / The shouldn't leave.
EX: He would not go. / He wouldn't go.
EX: It will not work. / It won't work. (will + not => won't)
As an adverb, "not" can also modify other adverbs, and adjectives:
EX: not all men dance
EX: not so handsome guy
"no" is an adjective. It pairs up with nouns. That is, it modifies a noun: