When I open a Keyword, there is Parent and Child Keywords, and also Related Keywords. What are they, what is the purpose of it, what is their behavior while publishing?
Taken from docs.sdl.com :
Select related Keywords to create non parent/child Keyword relationships such as see also links and related topics. For example, you can relate a Holiday Type Keyword to Sports Equipment Keyword.
Meaning that you can link Keywords which aren't from the same Category but can be related somehow. Whilst publishing this is sent as Metadata and is updated in the Broker, so you use the CD API to get those related Keywords of a given Keyword object (for example to build some dynamic functionality, criteria, etc.)
Of course you can get this in TOM.NET as well and include data from them in your publishing process, like creating links, etc.