Depending on your nationality, you may need a visa for Switzerland. The Student Exchange Office of ETH Zurich provides assistance with visa applications. More details will be sent to you once you have been admitted for an exchange at ETH Zurich. Do not enter Switzerland with a tourist visa: these allow you to travel to Switzerland as a tourist, but not to study there!
Find your country of citizenship on this website to determine if you need a visa (https://www.sem.admin.ch/sem/en/home/publiservice/weisungen-kreisschreiben/visa/liste1_staatsangehoerigkeit/u.html).
US passport holders need a visa to reside in Switzerland for more than 90 days. https://www.sem.admin.ch/sem/en/home/publiservice/weisungen-kreisschreiben/visa/liste1_staatsangehoerigkeit/u.html
Note: passport scans uploaded during the application must be valid for at least three months beyond the official end of the semester, incl. session examinations.
Here you can find the 4 Swiss visa desks in the US, where students have to arrange the pick-up of their Swiss visa (long-term stay over 90 days; no visa fee) after you received the visa authorization from ETH. The pick-up can be arranged via registered mail (most likely from the New York Consulate). In the very rare case, if you would need to travel to pick up the Swiss visa from New York - depending on your financial aid package, it can cover some of these travel costs.
Once you have entered Switzerland, you will have to apply for a residence permit with the appropriate authorities within 14 days of your arrival in Switzerland. You will receive more information about this procedure from the ETH Zurich admission letter.
In addition, health insurance is mandatory in Switzerland (and must be purchased on top of the insurance MISTI provides). ETH Zurich does not provide any health insurance. Usually, oversea students cannot be exempted and will have to take out a Swiss health insurance of their choice. Special offers for exchange students are available and information will be provided upon arrival at ETH Zurich. This cost is also reflected in financial aid packages.