This article from 'The Garden of Phrases at Grammar.ccc.com gives a good overview:
ABSOLUTE PHRASE
Usually (but not always, as we shall see), an absolute phrase (also
called a nominative absolute) is a group of words consisting of a noun
or pronoun and a participle as well as any related modifiers. Absolute
phrases do not directly connect to or modify any specific word in the
rest of the sentence; instead, they modify the entire sentence, adding
information. They are always treated as parenthetical elements and are
set off from the rest of the sentence with a comma or a pair of commas
(sometimes by a dash or pair of dashes). Notice that absolute phrases
contain a subject (which is often modified by a participle), but not a
true finite verb.
Their reputation as winners secured by victory, the New York Liberty
charged into the semifinals.
The season nearly finished, Rebecca Lobo and Sophie Witherspoon
emerged as true leaders.
The two superstars signed autographs into the night, their faces
beaming happily.
When the participle of an absolute phrase is a form of to be, such
as being or having been, the participle is often left out but
understood.
The season [being] over, they were mobbed by fans in Times Square.
[Having been] Stars all their adult lives, they seemed used to the
attention.
[[Being] A scientist, he always thinks about what is best for the
people.]
The dragon [having been] slain, the knight could rest.
Another kind of absolute phrase is found after a modified noun; it
adds a focusing detail or point of focus to the idea of the main
clause. This kind of absolute phrase can take the form of a
prepositional phrase, an adjective phrase, or a noun phrase.
The old firefighter stood over the smoking ruins, his senses alert to
any sign of another flare-up.
His subordinates, their faces sweat-streaked and smudged with ash,
leaned heavily against the firetruck.
They knew all too well how all their hard work could be undone – in an
instant.
It is not unusual for the information supplied in the absolute phrase
to be the most important element in the sentence. In fact, in
descriptive prose, the telling details will often be wrapped into a
sentence in the form of an absolute phrase:
Coach Nykesha strolled onto the court, her arms akimbo and a large
silver whistle clenched between her teeth.
The new recruits stood in one corner of the gym, their uniforms stiff
and ill fitting, their faces betraying their anxiety.
A noun phrase can also exist as an absolute phrase:
Your best friends, where are they now, when you need them?
And then there was my best friend Sally – the dear girl – who has
certainly fallen on hard times.
I think it's best to disambiguate 'nominative absolute' and 'absolute construction' (I'd see the latter term as a hypernym). Constructions with free-standing adjectives/adjectivals:
Desperate to get to the car, he ran without an umbrella over his head.
Exhausted, he gave up after 20 miles.
are certainly recognised as 'absolute constructions' by many authorities.
