How frequently is the word 'Appropriation' used in American English? In what contexts might young people commonly hear it?
Today appropriation is probably most often to be found in the context of critical and cultural studies, designating an artist's conversion of established artistic elements to new uses.
Before the 1980s, however, the word was usually used of money: either the conversion of someone else's money to one's own use, or the act of a legislature assigning revenue to specific uses or categories of use. Those uses are still very much alive.
"Young people" covers a lot of territory; but I imagine that primary- and secondary-school students would only encounter this word in the latter sense. And they'd be most likely to encounter it (and the verb appropriate) colloquially as an ironic euphemism for theft.
As for "how often" it's used: see Andrew Leach's Google Ngram. It's not rare, but it's not a term that comes up often in conversation. It's probably used as-often-as-it-is-needed-and-a-little-bit-more.
Young people who follow domestic U.S. politics will almost certainly have heard or read the term appropriation in connection with the two very powerful Committees on Appropriations of the U.S. Congress (one in the House of Representatives and the other in the Senate). As the Wikipedia article on the House Appropriations Committee points out,
The constitutional basis for the [House] Appropriations Committee comes from Article one, Section nine, Clause seven of the U.S. Constitution, which says
No money shall be drawn from the treasury, but in consequence of appropriations made by law; and a regular statement and account of receipts and expenditures of all public money shall be published from time to time.
The wording of the U.S. Constitution was adopted by the Constitutional Convention in 1787. Nineteen years later, Noah Webster's Compendious Dictionary of the English Language (1806) offered this definition of the verb appropriate:
to set apart for a certain purpose, or for one's self
In contrast, Merriam-Webster's Eleventh Collegiate Dictionary (2003) lists three distinct definitions of the verb:
1: to take exclusive possession of: annex 2: to set apart for or assign to a particular purpose or use 3: to take or make use of without authority or right
Definition 3 of the verb appropriate expresses a meaning almost diametrically opposed to that of the older definition 2: The appropriation it describes is more akin to plundering than to lawful requisitioning or allocation. Nevertheless, use of the verb appropriate in the definition 3 sense is fairly common in the United States today, as is use of the noun appropriation based on the definition 3 verb. But whether members of a particular group of young people understand it in that way (or at all) is another question entirely.