Two different relational databases.
Your task is to write a code to transfer the data from the first database to the second database.
Some tables in the database you are transferring to are of the same structure as the table you are transferring from, the transfer of these tables is as simple as "INSERT INTO DbA.TableA (...) VALUES SELECT * FROM DbB.TableB".
Some tables in the database you are transferring to have different structures and different purposes. After proper analysis, you understand the relations and you understand the right transformation you need to code.
My question is: how do you express such knowledge? How do you express the transformational relations between two databases? Are there any tools or diagrams?
The best way I know right now is writting the list of tables of the first database and for each table describing how it is to be transformed into the second database. Is it possible to make this more formal/concise/cool?
If you are wanting a toolset and work in the Microsoft database stack then this is exactly what SQL Server Integration Services (or SSIS) is used for.
If you are wanting to document the process then you would typically write an interface definition document (IDD). There are many examples on Google but here is something to get you started.