I'm not an expert in baby-naming, so I hope other people would post better answers (or correct me if I'm wrong!!)
- A character shared between siblings is called a 돌림자, namely "Shared Character", or 항렬자, meaning "Generation Character".
Before I can explain how 돌림자 works, I should first briefly talk about how families work in Korea.
You may have noticed how there are so many Korean people whose surnames are Kim (김). Strictly speaking however, their surnames are not one single kind of 김. In Korea, there are dozens of different "family clans" (which are called 본관), each of which has its place of origin, and a family name. So even if two people have the same family name, 김 for example, if they do not have the same place of origin, then they are not really from the same family. (More on this: Wikipedia)
Within a family clan, each generation (called 항렬) shares one 돌림자. So a 돌림자 is not only shared between siblings, but between the entire generation of a family clan. And each family clan has its own system for 돌림자 (systems typically follow the principles of Wu Xing, AFAIK.)
My 돌림자, for example is 행(行), and it's also supposed to be the last character of the name, according to the 돌림자 system my family clan has. And if somebody in my family clan recognizes that the last character of my given name is 행, while my last name is 이, they would suspect that we belong to the same family clan, and ask me if that's the case, like "혹시 XX 이씨 아니세요?", where XX denotes my family's place of origin. (This happens quite a lot. :D)
- A given name for a baby is decided based on 사주팔자.
사주팔자 is basically a kind of an ancient astrology. I don't know much about it. People often consult 작명가 or maybe 스님 (a Buddhist monk) in order to name their babies. Not many ordinary Korean folks like me have an extensive knowledge on 사주팔자, in my opinion.
But many people nowadays also decide to give Korean names to their babies (names with no Chinese characters), so in that case, 사주팔자 is less of a concern.