Why is the total wealth of the world not a conserved quantity? I have read that the GDP of the world is growing. I do not understand how this is possible. If I make profits out of a commodity or a service, somebody pays me for it. Hence, their wealth has to drop. So, shouldn't wealth only get redistributed? If so, how is it possible for the world GDP to grow?
There's no such thing as a law of conservation of wealth. Wealth is created and destroyed in many ways. It is created when society produces goods and saves them. It is destroyed in disasters, or consumed when we use up our savings.It is created when we create knowledge, software, etc.

In a toy example: when a society starts out without any saved food, it produces food, eats half and saves half, then it is richer at the end. The accumulated food is an asset. 

There's all kinds of wealth in a society: patents, ideas, buildings, roads, cars, computers, education, etc.

What does not get created or destroyed are nominal claims that we have against each other: if I give you a paper promising to pay you 10, that makes you 10 richer and it makes me 10 poorer but that's it. This means that its pretty had to figure out what is a society's total wealth, because there's all kinds of claims, explicit or implicit that are hard to value.