Twitch recommends having at least an Intel Core i5-4670 processor (or its AMD equivalent), 8GB of RAM and Windows 7 or newer. (Don't worry; you can stream from a Mac, too.)
If you're streaming PC games, you'll need a graphics card strong enough to support whatever you're playing and, ideally, one that supports DirectX 10 and up. The faster your Internet connection, the better — you should probably aim to have an upload speed of at least 3MB per second, which should be attainable on most home Internet connections or even via mobile.