[grammatic question] What is "a God"? What does this phrase mean?

1) As far as I know, the word "God" is monotheistic.

True.

"God" - capitalized but not at the beginning of a sentence - is being used as a title and can be accurately defined as "the god".

2) In contrast, "a god" is not (it is probably polytheistic.)

Not necessarily!

The term "a god" recognizes at least belief (present OR past) in deity or deities

but is not necessarily polytheistic.

FOR EXAMPLE: if I say, "Thor was a Norse god and Zeus was a Greek god"

does that mean that I am polytheistic? In fact, "poly-religionist"? No, of course not. It only means that I recognize that these names are or were assigned to deities by people.

When non-polytheists speak of "a god" we are normally speaking of deities that are or were recognized by deities by someone at some time...and NOT speaking of deities that we ourselves believe to be real and existent deities.

It's just part of our language

an imprecision of our language, if you will

and

it was also (clearly, if one reads the Bible) part of the ancient Hebrew language.

3) However, that phrase, "a God", has both the indefinite article ("a") and the capital letter ("G"). Even in the Bible, God says "I, the LORD your God, am a jealous God". Is "a God" totally different from "a god"? Is this something like "he is an Einstein"?

No.

This is a monotheistic usage of the typical term "a god". In this case the speaker is recognizing that other deities are believed to exist but that only one of them (capital "G" God) is real and existent. The monotheistic speaker is (either explicitly or by implication) comparing their "God" to other gods (gods that they believe to be something other than existent and/or something other than actual deities).

Gen 1:1 In the beginning H7225 God

first time in the bible you read this.

what does it mean?

how do you spell it out? (in Hebrew)

אֱלֹהִים ʼĕlôhîym, el-o-heem'; plural of H433; gods in the ordinary sense; but specifically used (in the plural thus, especially with the article) of the supreme God; occasionally applied by way of deference to magistrates; and sometimes as a superlative:—angels, × exceeding, God (gods) (-dess, -ly), × (very) great, judges, × mighty.

this actually means GOD and can included HIS CREATIONS

(better said CHILDREN).

Gen 1:26-27? ALMIGHTY GOD was not talking to HIMSELF.

supreme GOD (or supreme RULER).

not a man ruler or king in some magistrate.

this word and name and title stems from this title.

אֱלוֹהַּ ʼĕlôwahh, el-o'-ah; rarely (shortened) אֱלֹהַּ ʼĕlôahh; probably prolonged (emphatic) from H410; a deity or the Deity:—God, god. See H430.

now what is a deity?

look it up.

- DIVINITY? (SUPERNATURAL TRAITS)

now the prime word: (used)

אֵל ʼêl, ale; shortened from H352; strength; as adjective, mighty; especially the Almighty (but used also of any deity):—God (god), × goodly, × great, idol, might(-y one), power, strong. Compare names in '-el.'

Ezekiel?

Daniel?

GOD'S TITLE and NAME is here.

- Israel?

the prince that prevails and rules with GOD

- is what this name and title means.

GOD is THE MOST HIGH

THE CREATOR

SUPREME RULER

HE is also DIVINE (and ETERNAL) in NATURE.

ALWAYS HAS BEEN?

ALWAYS SHALL BE?

I am sure this answers your question.

more study is suggested to bring out even more truth (wisdom).

I am a Christian, and I sometimes hear this phrase used during religious discussion with other Christians. In that context, what Christians mean by "a god" or "a God" is any person or thing, real or imagined, past or present, that is not the monotheistic God of the Abrahamic religions, and which is either worshipped by people or is put in such a position of importance in a person's life that the monotheistic God takes second place to it.

Some examples:

A stone figure that is put into a temple and prayed to or that people bring offerings to.

The Greek and Roman Gods which old stories exist about that were once sincerely worshiped and now are only read about in mythology books.

Any life goal which so consumes the mind of the person who wants it, that they are willing to do anything or sacrifice any relationship to see it come true.

An abstract idea like being rich or getting as much great sex as possible, which is the most important thing to a person, is "a god" to that person.

This is a concept that Christians share in religious conversation. I do not see these last two examples used in the whole English speaking culture.