What do you think is the most important role of the UN in the world today?
The most important role of the UN today is as a moral force, both in focusing relief and attention on humanitarian issues (e.g., refugees needing sanctuary) and in fostering and brokering peace among nations. The UN, founded in 1945 after two very bloody and costly world wars, takes seriously its role as peacemaker and as an arbiter of international conflicts. Today, almost all the nations in the world are a part of it. The UN cannot stop a war if a state is determined to become aggressive, but it is so respected and reputable (even if not spotless) that its words, and especially its condemnations, carry weight. For example, the US sought the approval of the UN Security Council before launching war against Iraq in 2003. It went ahead without that approval, but the fact that it wanted it shows the UN's moral power.
One part of the UN is its judicial branch, the International Criminal Court. While countries of the global south, particularly in Africa, have protested that it is harsher on poorer nations, it has played an important role in investigating and trying crimes against humanity. It thus provides a moral yardstick for what constitutes acceptable nation-state behavior.
In my view, the most important role of the United Nations (UN) in our modern world is the same as it has always been. Its most important role is to make it easier for countries to make contacts with one another so that those countries can be more likely to cooperate.
One theory of international relations (called idealism or liberalism) holds that countries will be less likely to come into conflict if they have more connections to one another. The UN allows countries to make these connections. It gives them a convenient way to cooperate with one another on things like health or ending poverty or preserving sites that are important to world cultural heritage. When countries come into contact with one another on a broad range of topics like these, they come to have more mutual understanding. When disputes emerge between them, they are less likely to escalate because the countries are used to dealing with issues diplomatically and because they have so many ties with one another. Providing the opportunities for such contacts is the most important role of the UN.