The basic meaning of "international" is "something that involves two or more different nations." It could be a business transaction, an agreement, or a shipment. It could involve the nations as nations--such as a treaty or a trade agreement. It could involve people living within the nations.
In the United States, a mutual fund that invests in "international stocks" means a fund that invests in stocks that are all outside the United States. Some tiny airports near the U.S.-Canadian border are "international airports" because planes fly from the U.S. airport to or from Canada.
The usual words for "applying to the whole world" are "global" or "worldwide." A statistic that is includes all countries in the world might be described as "global." For example, WHO says "71.4 years was the average life expectancy at birth of the global population in 2015."
NATO is an international organization, because it involves many different nations. It is not a global organization, because it does not involve every (or almost every) nation on the world. The United Nations is both an international and a global organization.
The basic meaning of "universal" is "everywhere." It has many different senses. A mixture of activated charcoal and other things is called the "universal antidote" because it is useful in counteracting almost any poison. Someone with Type O negative blood is a "universal donor" because this blood can be safely used to give a transfusion to someone of any blood type. People say "smiles are the universal language" meaning that smiles signal friendliness in all countries and all cultures.
Sometimes "universal" can mean "global;" for example, "the 1956 Winter Olympics received universal acclaim." "Universal" means "by everyone" and in this case, "everyone" meant "the whole world.
"Universal" often refers to something grand, like the cosmos; or, sometimes, religion or morality. We speak of the "universal laws of physics."