By default your streams/broadcasts on Twitch.tv will not be saved as videos and available for watching later. You can change that if you wish.
Past streams are named as VOD (Video on Demand) on Twitch. There are also highlights which you can make from your broadcasts and they’re saved as a new video and also can be exported to Youtube. And finally there are clips which you can “clip” from any Twitch stream, not only your own, but they cannot be exported nor saved and they expire if the source video expires.
Go to your Twitch.tv dashboard page: https://www.twitch.tv/dashboard -> Settings -> Channel. There’s an option “Store Past Broadcasts”, which automatically saves broadcasts for up to 14 days (60 days for Partners, Turbo and Prime users). Check it.
That’s pretty much it. Remember though that past broadcasts will be saved only for 14 days for regular streamers, 60 days for turbo subs and for partners. But, the highlights which you can make from your past streams will be saved indefinitely for everyone. So it’s worth creating highlights if you wish to save your broadcasts for as long as Twitch will be available on the Internet.