What can I make with Twine?
At its heart, Twine is a tool for creating hypertext.
The difference between hypertext and a linear story, the kind found in books and magazines, is that it allows the reader to have some measure of control. In other words, the reader has some ability over what they interact with next. In a story about a haunted house, for example, the reader might be able to tell the protagonist to "Turn around and run" or "Venture deeper into the mausoleum". In a nonfiction piece, the reader might ask to learn more about my aunt, who went missing.
The convention that has emerged over the past three decades is that readers navigate hypertexts by clicking links. In this sense, you're already a seasoned hypertext reader by reading this very page! You clicked several links to reach this text, after all, and you've probably clicked an uncountable number of links in your life so far.
Because hypertext branches so much, it's easy to get lost in your own work. Much of Twine is dedicated to helping you keep track of your work's structure visually with a Passages View, so you can see what your readers' experience will be like.
Can I build games with Twine?
However, things a little more complicated than they initial appear. Twine itself can be thought of as more of an editor that helps package up Stories. What provides the underlining conditional logic, variables, and other trappings of game programming are Story Formats.