Internationalism is defined more or less (there are different definitions, with different nuances) as the idea that countries should work together to solve and prevent problems that arise in the world. This idea holds that countries should not be allowed to just do whatever they want and to act on their own when their actions will impact the whole world.
This idea is important because it leads to a different type of world order. It leads to one in which there are many international organizations like the ones we have today. These reduce the freedom of countries to do what they want. But, hopefully, they also improve the general state of the world.
There is no way to know where the idea originated. In the modern world, the first major attempt to bring it about came with the creation of the League of Nations after WWI. The idea has taken hold better after WWII and has led to the creation of things like the UN, the IMF, the WTO and the World Bank, among many others.
The practice of maintaining cordial and friendly relations on all fronts especially political and economic between different countries for mutual benefit is known as "Internationalism." Cooperation is the key element of internationalism. It implies a mutual respect for one another's culture and emphasizes peaceful co-existence amongst the various nations.
It would be very difficult to say precisely when 'internationalism' began although the race amongst the European nations to acquire overseas colonies in the sixteenth centuries would be a probable starting point.
The UN is the most important institution which fosters the growth and spread of internationalism by encouraging peaceful cooperation amongst the various nations of the world.