Typically NGOs are using free or low cost software, but this market is broad. In my environment the cloud based Smart Sheet is often used, which is not free but affordable.For bigger projects the MSProject is popular, because MS offers it in low cost modelts for non-profit NGOs.
I agree with Ferenc. EXCEL also can be used to track schedule and cost such as planned timeline vs. actual timeline and planned cost vs.actual cost. Earn Value Management is the key for project monitoring and controlling and the formulaes can easily be used in EXCEL.