Interactive fiction allows the reader to make choices that take them down different narrative paths. The classic example in print literature is the Choose Your Own Adventure Books published by Bantam Books in the 80s and 90s. There are also many, many examples of digital interactive fiction where the reader clicks on a choice or has a limited number of commands that they can type in to move the narrative forward. The classic example here is Colossal Cave Adventure, a text adventure game written by Will Crowther.
The line between story and game is often blurred in interactive fiction, perhaps because the reader becomes the main character and a positive resolution of the plot is akin to winning a game.
Click the Twine image on the left to "play" a Twine story about...Twine!
When using Twine with students, you may want to stick to just text. The power of the tool is in providing the interactive links that allow the reader to make choices. If you want to "level up" and include images, you'll need to figure out how to host them.