Each and each of

Each is a determiner. It is used before a singular noun.

Each boy wore a hat. (NOT Each boys wore a hat.)
Each person is unique.
While we were on a holiday we enjoyed each moment.

Each of

Each of is used before a pronoun or a noun with a determiner (e.g. demonstratives, articles and possessives). It is followed by a plural noun. The verb is singular but can be plural in an informal style. Note that we use object pronouns (e.g. us, them) after each of.

I have invited each of my friends. (NOT I have invited each my friends.)
Each of us has problems.
Each of us have problems. (more informal)

Position of each

When each refers to the subject it can go with the verb. Different structures are possible.

Subject + auxiliary verb + each + other verb(s)

They have each been invited. (= Each of them have been invited.)
They were each invited. (= Each of them were invited.)

Subject + each + main verb

We each think differently. (= Each of us think differently.)

Position with object

Each can follow an object.

She kissed them each on the forehead. (= She kissed each of them on the forehead.)
She bought us each an ice-cream.

Pronouns referring back to each can be singular (more formal) or plural (more informal).

Each girl gave her version of the story.
Each student was asked to bring his / her identity card. (More formal)
Each student was asked to bring their identity card. (More informal)