How MANY people ARE there in your family? There ARE five people in my family. 'People' is the plural of 'person', so it's countable. One person ---> Two people. If you think of 'people' as being 'personas' (plural) in Spanish, not 'gente' (singular), it may help you to remember that 'people' is countable and plural. NB The word 'persons' does exist, but it is mainly used in formal notices and legal documents - not in everyday English.

In most contexts, it's plural and countable: "There are fifteen people in the room." An effective singular is person: "There is one person in the room". ("Person" has its proper plural persons, but it sounds official or stilted: "Authorized persons only", "Persons tampering with the smoke alarms shall be subject to prosecution", ...) There is another use of "people", which can take the plural "peoples": "The peoples of the world [which might otherwise be considered distinct] are united in their love of music". But this is a bit specialized.

My English is not well but I am a maths teacher. People are never uncountable. "How many people are there?" Is correct.