with thanks to Herr Gilleland

With thanks to Dr. Mike Gilleland, I've learned a new grammatical term that describes a familiar turn of phrase:

nominative absolute

A nominative absolute is a phrase that modifies an entire clause while standing apart from it. (The standing-apart is what makes it "absolute," grammatically speaking. I'm still not sure what makes it "nominative" aside from the idea that, if the expression modifies a clause, then the subject of the clause must necessarily be in the nominative, i.e., subjective, case.*)

Wikipedia offers good examples of the nominative absolute, plus a second, contrastive set of sentences that are either complex or showing adverbial prepositional phrases.

The dragon slain, the knight took his rest.
The battle over, the soldiers trudged back to the camp.
The truck finally loaded, they said goodbye to their neighbors and drove off.
We sit side by side, our legs touching, comfortable in the warm silence our two bodies create.
Spring advancing, the swallows arrived.

Clauses or adverbial prepositional phrases:

Because the dragon was slain, the knight took his rest.
When the battle was over, the soldiers trudged back to the camp.
After the truck was finally loaded, they said goodbye to their neighbors and drove off.
With our legs touching, we sit side by side, comfortable in the warm silence our two bodies create.
When spring was advancing, the swallows arrived.

Other examples of the nominative absolute:

Weather permitting, we will have a barbecue tomorrow.
All things considered, it's not a bad idea.
This being the case, let us go.
The referee having finally arrived, the game began.

As you see, a nominative-absolute phrase contains a noun (or noun phrase) plus a participle.

weather (noun) + permitting (present participle)
all things (noun phrase) + considered (past participle)

Note that the nominative absolute looks similar to, but isn't the same as, a participial phrase used as a modifier.

Sitting on the branch, the owl stared imperiously downwards.

Why isn't Sitting on the branch a nominative absolute? Because as I said above, a nominative absolute contains a noun plus a participle—in that order. Yes, there is a noun in Sitting on the branch, but (1) it comes last, and (2) it's part of a prepositional phrase, which doesn't have the grammatical value of a noun (it's adverbial in this context, modifying Sitting).

In the comments, try making your own sentences with nominative absolutes.

His mission completed, Jensen left the diarrhea-soaked mattress for others to clean.
Her heart shattered, Belinda knew she had to look elsewhere for proper tentacle sex as the disgusted octopus slid silently beneath the waves.

Pulse pounding, His Holiness opened the door to the children's play room. 

__________

*In case you're not familiar with cases, which cover the functions of nouns, noun phrases, pronouns, and even adjectives, they go something like this (if I may use German as a template for English—Nominativ, Akkusativ, Dativ, Genitiv):

nominative = subject → e.g., He in He farted lustily.
accusative = direct object → e.g., the fart in She smelled the fart.
dative = indirect object → e.g., at him in She farted at him in kind.
genitive = possessive → e.g., his (fart) in But his fart proved far more powerful than hers.

So by the above reckoning, he would be the nominative case; him would be the accusative case; to him (or even arguably for him) would be the dative case; his [+ noun] would be the genitive case. Look at this sentence:

Spencer gave Max's old ball to the new dog. → He gave it to him.

blue = nominative (Spencer = the subject of the sentence)

red = genitive (Max's is in the possessive case)

green = accusative (old ball is the direct object of gave)

purple(ish) = dative (to the new dog → the new dog is the indirect object of gave, and to is the marker indicating this is an indirect object)

The prepositions for, at, to, etc., aren't always there to let you know something is in the dative case. In such instances, you can normally use logic to deduce cases.

Fran told us a scary story.

Fran = nominative case (subject of the sentence)

a scary story = accusative case (direct object of told)

us = dative case (indirect object of told)

Maybe I should step back and ask whether you know the difference between direct and indirect objects. Whatever "receives" the action of the verb first (logically or chronologically) is the direct object. Whatever receives the action after that is the indirect object.

I kicked the ball at Cindy.

The action is kicked. What receives the action of kicking first? Obviously, it's the ball, which receives the action before Cindy is hit by the kicked ball. Cindy then receives the action of kicking indirectly: she doesn't feel the direct force of the kick, but she feels the kick's force indirectly through the impact of the ball. Does that make sense?