A public nuisance was someone who interfered with the rights of the public.
In 2373, after hearing Jake Sisko and Nog's claims about Elias Giger, Odo threatened to charge them with "filing false reports, obstructing justice and being public nuisances". Nog worried that the charges would look bad on his Starfleet Academy record, but Jake assured him the security chief was just trying to scare them. (DS9: "In the Cards")