Is the United Nations a relevant and effective organization in the modern world?
The UN is not relevant and effective if you expect it to be able to actually prevent all conflict or force countries to live up to treaties. This is because it lacks the power to do anything unless the Security Council agrees unanimously. This means that it is generally helpless to do things like preventing Iran from getting nuclear weapons or stopping the violence in Syria.
However, it is relevant and effective if you see it more as a force for humanitarianism and improving the lives of people around the world. The UN sponsors groups like UNICEF that help children around the world. It helps in places like Haiti where political chaos threatens people. It cannot help everyone everywhere, but it is relevant to many people.