I read the following definition for accusative absolute, but the many syntactical terms (based on Latin) confound me: accusative and nominative absolute.
a construction in English, especially colloquial English, consisting of a pronoun in the accusative case joined with a predicate that does not include a finite verb and otherwise identical with the nominative absolute (as him being my friend in “him being my friend, I granted his request”)
I first heard this at 54:31 in this Youtube video.
The accusative is a noun case in various languages including German and Latin and Greek; as applied to English the term denotes the objective case, shown in personal pronouns me, him, her, us, and them (in contradistinction to the subjective case, sometimes termed nominative, of I, he, she, we, and they).
An absolute construction is a way of tying an extra statement to a sentence while remaining rather vague about its relation with the statement made by the main clause. The grammatical transformation involves substituting a participle (in English characteristically the present participle, the -ing form) for the main verb of the extra statement:
Helen assigned the task to Rosa. [this will be the main clause]
John was away from the office. [this is the extra statement]
John being away from the office, Helen assigned the task to Rosa.
Helen assigned the task to Rosa, John being away from the office.
We don’t know if Helen is filling in for the absent John as boss, or if she is always boss but would have assigned this task to John had he been around, or just what if anything would have happened differently had he been around.
Another example is the much-disputed Second Amendment to the U.S. Constitution (here with slightly modernized punctuation):
A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
Here it is a matter of long dispute whether the absolute, here the italicized portion, merely justifies what the main clause decrees [because a well regulated militia is necessary] or limits its application [insofar as a well regulated militia is necessary].
Sometimes the actual present participle can be omitted as understood:
My wife came out to me in the garden, her face [being] a mask of grief.
In none of these examples can we see the case of the absolute’s grammatical subject, whether it be subjective or objective. For in English, case becomes manifest only with personal pronouns. So now let us revisit the first example, supposing a context that is largely about John, so that instead of using his name we now use he or him.
He being out of the office, Helen assigned the task to Rosa.
Him being out of the office, Helen assigned the task to Rosa.
The first version, using the subjective pronoun He, exemplifies what is sometimes termed a nominative absolute. The second version, using the objective pronoun Him, exemplifies what is sometimes termed an accusative absolute.
Although the accusative absolute construction is eminently grammatical in classical Greek, the second version above is not really suitable for formal register in English, as your own source has noted. If it appears in a context where formal register is expected and demanded, it will strike a lot of editors and professors as semi-literate, in the same way as “between he and I” or “Me and my friends want . . .”—violating normative expectations of which pronoun case is appropriate to the positions object of preposition or subject of clause. The position of subject of absolute construction is likewise associated specifically with the subjective case, if it is to be filled by a personal pronoun in formal register.
The distinction between “nominative absolute” and “accusative absolute” in English, such as it is, completely disappears when that subject position is not occupied by any of the personal pronouns that show case (as “you” does not). Since in the video linked by OP Mr. Justice Scalia is talking about the second amendment, which as seen above contains no personal pronoun anywhere, it makes no real sense for him to characterize the absolute construction there as a specifically “accusative” one. But he ventures it only as a casual opinion. He is perhaps hoping to intimidate people with less grammatical knowledge, while being sufficiently noncommittal to stay out of trouble in case someone questions him who has more grammatical knowledge.
Accusative case, for the most part, is a fancy way of saying objective case. So your example was:
a construction in English, especially colloquial English, consisting of a pronoun in the accusative case joined with a predicate that does not include a finite verb and otherwise identical with the nominative absolute (as him being my friend in “him being my friend, I granted his request”)
"A nominative absolute is a noun phrase that begins or ends a sentence. The phrase has no grammatical connection with the rest of the sentence. Most nominative absolutes contain a participle or participial phrase which modifies the noun or pronoun."
Examples: "A well regulated Militia being necessary to the security of a free State, the right of the people to keep and bear Arms shall not be infringed." From here
and
"The weather being rainy, we decided to postpone the trip." From the same source as the definition.
It is essentially what is called absolute phrase, a phrase that functions outside of the main sentence but also provides additional information.
[Him (accusative pronoun) being (nonfinite verb) my friend (object/complement of nonfinite verb)] nominative absolute, I granted his request.
The reason why it might be called an accusative-abosulte is that Him is an objective case pronoun.
First, there is no such thing as "accusative-absolute", as a general grammatical term.
All grammatical terms are relative to individual languages. This term is relative to Greek and Latin.
But not to English -- English has no accusative case. So if what you're reading is sposta be about English grammar, you should automatically distrust it.
Second, the terms accusative, finite verb and nominative absolute.
Brian's answer deals with accusative case and absolutive constructions.
Nominative is another Latin, Greek, and German case; English nouns don't have cases.
That leaves finite verb.
Any English verb (well, almost any -- this doesn't apply to modal auxiliaries) can be used two ways:
They have very different kinds of syntax. Finite verbs can be main verbs, but non-finite verbs are always in subordinate clauses. Only finite verbs have tense, and one is required in every main clause; but you can't mark tense on non-finite verbs -- *You wanted to went yesterday is ungrammatical.
