Every one

We use every one of before a pronoun or a determiner (the, my this etc).

Every one of the children was happy. (NOT Every one the children was happy.) (NOT Every of the children was happy.)

Everybody and everyone

The expressions everybody, everyone, everything and everywhere are indefinite pronouns. They are used with singular verbs.

Everybody has agreed to come. (NOT Everybody have agreed to come.)
Everything he does is either illegal or immoral.

Pronouns used to refer back to everybody / everyone can be singular or plural. Singular forms are preferred in a formal style. Plural forms are preferred in a less formal style.

Has everybody brought his or her book? (Formal)
Has everybody brought their books? (Less formal)

Notes

There is a difference between every one and everyone.

Everyone can only refer to people. Every one can refer to things as well as people.

Every one of his books is worth reading. (NOT Everyone of his books is worth reading.)
Every one of us can swim. (NOT Everyone of us can swim.)
Every one of us is excited about the offer. (NOT Everyone of us is excited about the offer.)

Everyone should know how to swim. OR Every one should know how to swim.