A few disclaimers before giving this a crack for you:

1. This is only for undergraduate international students (I think the decision varies a lot being international because the reputation of the university becomes a lot more relevant if you plan to head back to work in your original country)
2. Rankings are fast approximations which are outclassed by good research. For specific subjects, the rank and order changes dramatically. I will answer this for your average student but it really varies tremendously if you are looking to study computer science or engineering or philosophy or folklore and mythology.
3. In choosing the university for me (I answered this here: Why choose Harvard over other universities?), I had to make this choice first hand out of Harvard, Stanford, Princeton, Yale, Columbia, Wharton, Cambridge etc so I put my money where my mouth is so to speak and followed my own ranking.
4. My ranking is slightly biased in favor of the universities that facilitate outperformance in finance/consulting/entrepreneurship/technology/law school/business school over academia.

With that out of the way:

1. Harvard
2. Stanford
3. MIT
4. Yale
5. Princeton
6. UPenn (Wharton)
7. Columbia
8. University of Cambridge
9. University of Oxford
10. California Institute of Technology
11. Brown
12. = UPenn (ex Wharton) + Duke
13. Williams
14. Ecole Hotelier du Lussaune (number 1 hospitality program in the world, great return on investment!)
15. Cornell University
16. Swarthmore College
17. UC-Berkeley
18. University of Washington - Seattle (2nd most recruited computer science program in the US/right next to Microsoft)
19. Reed College (one of the more quirky options but 2nd most Rhodes Scholars of any liberal arts school and a curriculum that de-emphasizes formal grading and has its own nuclear reactor for undergrads!)