Taut means tight rather than slack. The tightrope ought to be taut and not dangling down by the lion cage.
It sounds like the word "taught" and means stretched tight, like a rope, muscles, or even nerves. It's nice to have a taut body with tight muscles, but not so great to have a taut mind — tightly wound and tense. "The Olympic gymnast's taut body was something she worked hard for, but she gave a taut reply to reporters who asked her so many questions about her past, which she wished to keep a secret." Taut is derived from an Old English word, meaning basically "to pull." If you're fishing, you're happy when your line becomes taut, because there's probably a fish pulling at it. Or an old boot.