A high level of English language is crucial for being able to do any kind of serious research at the US universities. Also, without a good knowledge of the language, it will be difficult for you to communicate with locals. This is why a test of English language is required, and for most, this will be TOEFL.
When you or your adviser are filling out the forms and arranging the paperwork for your visit, be careful about the Fulbright J-1 visa. My DS-2019 lists “student non-degree” under the “Exchange Visitor Category”, which caused some trouble at my host University. This fact led to quite a confusion, since for me to stay in USA with this status, my host institution had to enlist me as a student.
After being accepted, there are multiple documents and forms you will have to take care of before actually starting your Fulbright year.