In English, "man" is also a general noun for the human race, short for "mankind" but with a distinct meaning. We use "mankind" to refer to the human condition; we use "man" as a sort of technical term for "human animals."
"Mankind must learn to live in peace." "Man has lived on earth for a long time."
Same in German, as in English:
die Person: the person
die Personen: the persons
die Leute: the people (no singular form)
die Menschen: the humans
der Mensch: man, or mankind (collective noun, different from "der Mann," which is one male human, or "die Männer," which is plural males or men)
Edited to add: der Volk: the people, as in a nationality or population or gathering crowd (which also have their own words)
The German plural "Personen," much like the English plural "persons," is naturally-generated but isn't used much outside of legal contexts: "The person or persons who committed the crime will be prosecuted." In both German and English, "persons" is not part of usual speech; the used plural is "people." So in both German and English, "the plural for person (persons)" is a "regular" plural, but "the plural for person (people)" is irregular.
"People" is not quite the plural of "person". You talk about "one person" and "some people". And "people" does (at least in the US) take a plural verb ("some people are ..."). But there exists a plural "persons" (which seems to be mainly used by lawyers), and there exists a plural "peoples" (which is rare, and you might only see in sociology books). So I would say that English speakers nearly always use the collective noun "people" instead of the plural "persons".
'People' is definitely not treated as a singular in English:
"THREE people WERE walking down the street" - the emphasised words cannot be used with singular nouns. Compare: *"Three person were walking down the street". You can replace 'person' with 'woman' or 'cat' or any other singular noun, and it would be equally ungrammatical in English.
The only time 'people' can be used as a singular is when it refers to a single (usually ethic, linguistic or national) group, as in "Bedouins are a traditionally nomadic people" or "We are a proud people", in which case it can even be pluralised: "The peoples of Europe speak mostly Indo-European languages"
"człowiek" - the basic, Nominative form of the word for "a human" (although in English a more natural translation is often "a man").
"człowiekiem" - its Instrumental form, used mostly in sentences like "He is a good man" = "On jest dobrym człowiekiem".
"ludzie" - the basic, nominative form for the plural "people".
"ludźmi" - its Instrumental form (Oni są ludźmi = They are people).