Do I need to buy a computer?
It's not necessary for you as a student to own a computer at Brown, even if you are a Computer Science concentrator. The Computer Science Department provides computing facilities that are adequate for all our courses. Many students actually prefer to use departmental facilities for their CS assignments, partly because of its proximity to teaching assistants, and partly because, by doing so, they are in social contact with all their classmates.
However, life is definitely easier if you do own a computer. If you choose to buy your own computer, whether it runs Windows, OS X, or Linux makes little difference, so follow your preferences. While many CS courses that have programming assignments use Linux, this shouldn't be your only criterion for choosing an OS. You'll be using your computer in all your courses, not just CS courses, and if necessary you can install Linux in a virtual host.
Whichever OS you choose, your computer should be reasonably powerful. A smartphone or tablet won't suffice, and the very cheapest netbooks may cause more frustration than you want to experience. On the other hand, you don't need a top-of-the-line laptop.
As on most campuses, WiFi is ubiquitous. Students make much use of the cloud (provided by Google Apps for Education) for email, storage, etc.