When you first open Twine, you're shown the story list. This shows all of the stories currently stored in Twine, with links underneath each to edit, play, and delete the story. In the upper-right corner of the list are buttons that allow sorting the list, either alphabetically by story name or by the last time the story was edited.
To create a new story, use the green + Story button on the right side of the page. You'll be prompted for a name and then taken to its story map.
To import a Twine story from a file, use the Import link on the right side. Right now, only published Twine 2 stories and archives can be imported. You cannot import a Twine 1 story, either in published .html or .tws source form, for now, nor can you import Twee source code.
The Archive link on the right creates a file containing all of your stories. You can use this as a backup, or give this archive to someone else to import. This archive contains HTML with data specific to Twine, so in a pinch you can also extract your story text from this archive file without any special tools.
On iOS devices, archive files are created as a ZIP archives, which can be sent to a storage app such as Dropbox or Google Drive. Due to a limitation in iOS, however, this file will always be named unknown.zip. You can, of course, rename the file once it appears in your file storage app.
The Formats link on the right allows you to manage story and proofing formats in Twine.
At the bottom of the right side of the story list is a gauge showing how much space your stories currently take up. Each web browser imposes a limit of how much a single web site or application can save. Most allow 5 megabytes of space, which doesn't sound like a lot, but since Twine stories are all text, will last longer than you might imagine. If you start to run out of space, one easy solution is to archive an old story to a file, then delete it.
Selecting the version number displayed in the lower-right corner will bring up development credits.