Is "people" a countable or uncountable noun?
It is a countable noun. See explanation for details.
A countable noun is a noun which can be "counted" (i. e. used with numbers). We can say "three people", "six people", so it is a countable noun.
If a noun is uncountable, it does not have plural form.
The word "people" is plural so it is countable.
For example "water" is uncountable, because you cannot say "five waters". To "count" uncountable nouns you have to add other words (like litre, glass, spoon) which can "measure" the amount of uncountable nouns.
You cannot say "five waters", but it is correct to say for example "five liters of water". In this example you do not use number with an uncountable noun "water", but with a countable "unit" ("liter")
The rule above can apply only to uncountable nouns describing real things. You cannot "count" abstract nouns (like love, justice, smell) in this way.