A Tornados badge is a badge worn by people who were fans of the Quidditch team Tutshill Tornados. It features a double gold 'T' against sky-blue background.
Cho Chang, a student at Hogwarts School of Witchcraft and Wizardry a year over Harry Potter, was a great fan of the Tornados. Ronald Weasley was an opponent of the team and began a discussion with Cho, asking her how long she had been a supporter of the team. He later explained that many people had just recently started wearing those badges, presumably because the Tornados were winning the league at the time.