How to create table mapping from old database to new database structure

"Map" - something that can describe how to "travel" from 'A' (old database) to 'B' (new database)

It simply means "for each column in the old database NAMES table work out where that data would be stored in the new database structure". For example if olddatabase.NAMES has columns

ID, 
Name, 
AddressLine1, 
AddressLine2, 
ZipCode

but the new database has two tables NAME and ADDRESS then the mapping might be

old.NAMES.ID            --> new.NAME.ID
old.NAMES.Name          --> new.NAME.Name
old.NAMES.AddressLine1  --> new.ADDRESS.AddressLine1
old.NAMES.AddressLine2  --> new.ADDRESS.AddressLine2
old.NAMES.ZipCode       --> new.ADDRESS.ZipCode
old.NAMES.ID            --> new.ADDRESS.NameFK

Exactly how these "rules" are described, defined or used depends rather on the mapping tool you will be using. You may also need to consider some data transformation e.g. [Name] might have to be split into [FirstName] and [LastName]

I would suggest searching google for the term "Database Mapping" or "Data mapping" for further detail and, depending on which version of SQL you are using, do some research into SSIS or DTS (for older versions of SQL)

By the way, your Chief Software Architect writes instructions as if they were a college lecturer - quite strange that :-)