Translation:What non-governmental organizations do you know?
In the UK we refer to "private sector" and "public sector" organisations - "government" and "non-governmental" sound out of date.
This usually works in the USA as well, although private sector generally implies business for profit. One would probably specify if it is distinctly a private non-profit organization. The phrase "non-governmental organization" sounds clunky and I don't know that I've ever actually heard it spoken outright like that. We pretty much always abbreviate it to NGO.