I wanted know, what does means these sentences below:
Nothing went right today. Nothing went right again.
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"Right" in this context would mean "as I planned/wanted it to", which they are deeming to be "correct" or, as they say, "right".
So the person is saying that nothing went as they planned it or as they wanted it to.
In some contexts it will mean that they planned a series of happenings, and nothing happened as planned. In other contexts it would mean that a series of unfortunate things happened, and what they're essentially doing is rationalising the idea that they had planned that it would be different, or at least would never have planned for things to have gone the way they did, even though no plans were actually made.