Showing two Turing machines decide different languages
Languages are just sets (of strings). As with any pair of sets, you show that they're different by showing that one set contains an element that's not in the other. Note that, in general, the problem of determining whether two Turing machines decide the same language is undecidable, so there's no algorithm you can use to generate such a string, in general. But, for any two given Turing machines, you can still try to figure it out.