Congratulations, OP Tipping. I think this is quite a toughie.
In default of Reb Wm, I’ll have a shot at at least a partial, “bare-bones” answer, though the subject cries out loud for a scholar - which I’m emphatically not.
The Hebrew term in the Commandment, rendered in the KJV as “in vain”, is the word lamed-shin-vav-aleph --- le-shav. It is still in common use in modern Hebrew, to mean what “in vain” means in modern English, i.e. “unavailingly” ("he searched in vain"). It’s also used elsewhere in the Bible in exactly that sense (Psalm 127)*, but in the present case, I think that the meaning in the Hebrew Commandment (and in the KJV translation) is subtly different, and broader. If I had to render the Biblical commandment in modern English, I might say “You are not to make improper or unnecessary use of the name of the Lord your God” (the NEB says “you shall not make wrong use......").
This raises a whole lot of questions - first of all, what is the name of “the Lord your God”?
The Hebrew Bible refers to God in several ways. Genesis begins: “In the beginning, Elohim created the heavens and the earth”. In Genesis 2:5, there is a change: God is referred to for the first time as YHVH Elohim, and later in the Bible, sometimes as YHVH, sometimes as Elohim, and sometimes as YHVH Elohim. When Moses before the burning bush (Ex 3:13) asks God: what shall he tell the people is God’s name, God replies (as languagehat reminded us in a recent thread): “I AM THAT I AM: say unto the children of Israel I AM hath sent me”. The First Commandment says “I am YHVH thy Elohim....thou shalt have no other Elohim before me”.
Since ancient times, names have been very important, especially names of gods, with all sorts of magical properties ("a name to conjure with"), often kept very secret, not to be revealed, and certainly not to be trifled with (Robert Graves discusses this subject at some length, I remember, in “The White Goddess”, my copy of which was “borrowed” some years ago, alas). The God of Moses clearly prefers to keep his exact name to himself, even from his most faithful followers. To Jews, YHVH is the nearest thing we know to God’s name, and the prohibition against using it “in vain” is taken very seriously indeed. Jews refer to God by a variety of titles and appellations. I think it’s not entirely clear, by the way whether “Elohim” is actually a name, or a title (same applies in English, to “God”, really, doesn’t it? There are other Gods, both in Hebrew and in English. Perhaps the capital G makes it specific). To be on the safe side, an observant Jew will say “Elokim” if he says the word at all - say, when reading from the Torah (YHVH will be read aloud as Adonai - “the Lord"). God is normally referred to in common speech obliquely, as “the Name” (ha-shem), or as “the Holy One, Blessed be He” (ha-kadosh-barukh-hu), for instance. That is how Jews understand the original intention of the commandment. Members of other faiths have their own interpretations. God only knows....
In ancient times, when the Temple stood in Jerusalem, the name YHVH was spoken aloud only by the High Priest, and only once a year: in the Holy of Holies, during Temple services on Yom Kippur, the Day of Atonement. When the Temple was destroyed, the practice ended. I don’t think anyone today even knows how YHVH was pronounced: “Yahweh” and “Jehovah” are only guesses.
All the above refers, of course, to the use of God’s name aloud and in public. Private communication between an individual and his/her God is nobody else’s business, and is obvously not easily subject to regulation.
(Note: I see wikipedia has a lot of stuff about this, by people who obviously know a lot more than I do. To anyone interested, I’d say “start with Yahweh")
* Sobiest: most of the 48 examples of “in vain” cited in the Blue Letter Bible, are translations of various other phrases, not of le-shav.