1. Did you see the "Amazing Dancer" last night?

[Why is the quotation mark used? Is "Amazing Dancer" a title of a show?]

Titles of poems, songs, and magazine articles should be in quotation marks. Titles of books, plays, movies, artwork, and other major works should be in italics or underlined.

However, these days I’ve noticed people using quotation marks because they can't use italics or underlining on most (or all?) social media websites.

Thank you.

1. Did you see the "Amazing Dancer" last night?

[Why is 'the' used here? Is 'Amazing Dancer' a play or a movie?]

I don't know what "Amazing Dancer" is! It sounds as though it's a movie or play, though, or maybe a TV program.

When I googled "Amazing Dancer," this is what I got!!
https://www.google.com/search?q=%22amazing+dancer%22&oq=%22amazing+dancer%22&aqs=chrome..69i57j0j69i61j0l3.4005j0j7&sourceid=chrome&ie=UTF-8

The quotation mark is used in this sentence because "Amazing Dancer" is being referred to as a specific title or name. In this case, it is possible that "Amazing Dancer" is the title of a show or a performance. It is common to use quotation marks to indicate that a specific word or phrase is being used as a title or as something that should be set apart from the rest of the sentence.