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Twine 2 is the most recent release series of Twine. If you've never used Twine before, this version is recommended.
Twine 1 was the original version of Twine, and is still available to download.
Story formats are separate from the Twine editor, and turn the story you design in Twine into a playable HTML file. They allow you to add conditional logic, display effects, and other interactivity. (How do I decide which story format is right for me?)
Harlowe is the default story format for Twine 2. It isn't available in Twine 1.
SugarCube is a popular story format that is a descendant of the original SugarCube format packaged with Twine 1. Twine 2 comes packaged with the latest in the SugarCube 1.x series, but there's also a 2.x series available that can be added manually.
Twine stories are published in HTML format, which means you can publish your files on any web server.