I think you're looking for parenthetical. A parenthetical statement or clause is one that is not essential to the main sentence (i.e. it can be contained in parentheses). The opposite would be non-parenthetical or some synonym of essential.
Dashes and parentheses mostly work the same. Parentheses whisper while dashes shout. Commas are somewhat in-between. All three can be used to set off parenthetical statements.
Both of your examples are parenthetical statements. If you remove them from the sentence, it does not significantly change the core meaning or intent. The only difference in the following sentences is the tone:
The president (and his assistant) traveled by private jet.
The president -- and his assistant -- traveled by private jet.
The president, and his assistant, traveled by private jet.
If the fact that the assistant also traveled is essential, then it's not parenthetical, and should not be contained in one of these parenthetical structures:
The president and his assistant always travel together by private jet.
You may also be thinking of restrictive and non-restrictive clauses. Non-restrictive clauses are bonus information that can usually be enclosed in some kind of parenthetical structure.
Kaylee (who just graduated from high school) is an accomplished figure skater.
Side note: Pairs of dashes are used to set apart parenthetical information. A single dash is a different animal, and is used to highlight a dramatic action or statement.
They saw them too late as their ship rounded the headland -- pirates!