1. That's an excellent question, and the answer will vary from person to person. My opinion is based on my knowledge of American literature (including plays), as well as the plays most commonly taught in high schools, colleges, and universities.
Here are my picks for the ten most influential American plays:
1. Death of a Salesman - Arthur Miller - 1949
2. A Streetcar Named Desire - Tennessee Williams - 1947
3. A Raisin in the Sun - Lorraine Hansberry - 1959
4. Angels in America - Tony Kushner - 1991
5. Long Day's Journey into Night - Eugene O'Neill - 1956
6. Our Town - Thornton Wilder - 1938
7. The Crucible - Arthur Miller - 1953
8. Who's Afraid of Virginia Woolf? - Edward Albee - 1962
9. The Glass Menagerie - Tennessee Williams - 1945
10. Fences - August Wilson - 1983
Any one of these plays is worthy of being called the best and most influential in American history, and there's at least a dozen more that would fit the bill.
As for the most prominent playwright, it's a close race between Arthur Miller and Tennessee Williams—with Miller pulling into the lead.