If you’ve done something erroneously, you’ve made a mistake or accidentally given out wrong information. If you erroneously accuse your brother of eating the last slice of your pecan pie, then you’re wrong: he didn’t do it.
Erroneously traces back to the Latin word errare, meaning “to stray, err.” If something’s done erroneously, you can think of it as straying from the right path and ending up going in the wrong direction. If you erroneously report to the IRS that you only made $500 last year and you really made $10,000, well, that’s a big mistake. If you do something erroneously, it’s typically by accident — you think it’s right but it isn’t.