What are the following words called: Am, Is, Are, Was, Were, Be, Being, Been?

The words you cited are all forms of the verb “be”, which is also known as a copula or linking verb.

The term auxiliary verb applies to verbs, such as forms of be, have, and do, that conjoin with another verb to add syntactic or semantic information, such as grammatical aspects like the progressive aspect or perfective aspect:

• progressive aspect: be + present participle (e.g. am walking)
• perfective aspect: have + past participle (e.g. have walked)

Verbs such as will and shall combine to indicate future tense or conditional tense.

And, to add to the terminology conundrum:

"to be" is either

• a copula verb: it asserts a property

  John is a teacher

  Peter is nice

• an auxiliary verb: it is required to encode, e.g., tense or voice

  Max has been beaten up by members of this gang

• a full-blown main verb: roughly meaning "to exist"

  To be or not to be: that is the question

These distinctions can become quite fuzzy. Consider:

There is a unicorn in the garden

Is this the "exist"-reading of the verb, or is it copula use? I currently have no definite answer for this.