Question: I try to help some people with basic English, and I'm not able to explain why "Everyone" takes a "s" with the simple present verb, and why "People" takes no "s".

I explained "Everyone" as a group, and group means a plural, but have a singular form. Example: a football team.

But with "people", my explanation get confusing. How to explain it, considering the students ask for a logic, if they are learning from a language with a very different logic.

In linguistics, how is it called?

PS: why there is no "indefinite pronouns" tags, nor "linguistics" tag?

Answer: 
"Everyone" is equivalent to "Every person". Although we are talking about multiple people, we are grammatically using a singular.

* Everyone (singular) **works** overtime today.
* Every person (singular) **works** overtime today.

"People", on the other hand, is the plural for "person". "Persons" *can be* correct in certain circumstances. But you'll want to use "people" in most cases.

* People (plural) **work** overtime today.

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There are ways that you can say the same thing about a group of people, but you can grammatically use the singular if you want to:

* The Coca Cola employees (plural) **work** overtime today.
* The Coca Cola staff (singular) **works** overtime today.
Almost all English plural nouns end in *s* or *es* but there are about 20 or so nouns with irregular plurals (that aren't stolen from Latin) - and *people* is one of them. So it's treated the same way as a noun with *s* or *es* on the end.

*Everyone* is really the two words "every one" run together. So it's really *one* modified by *every*. You would use singular verb forms if *one* was the subject, and continue to do so if *one* was modified by another word.

Similar thread of logic with *everything*, *everywhere* and *everybody* (though "every body" does mean something different than "everybody" and "every where" doesn't work.)