If something is ongoing, it is happening now. If you have an ongoing argument with your neighbor, it means you were arguing before, you’re arguing now, and it looks like you'll be arguing into the future.
If something is ongoing, it is currently in action. Often in news reports, reference will be made to an ongoing investigation because the police are in the midst of it at the moment of reporting. Ongoing reports are currently in progress as are ongoing repairs. If your teachers say that there will be ongoing testing throughout the year, you won't be done with it until you get out of school.