Well-versed is used to describe someone very familiar with a specific subject, especially one that requires a lot of experience or study. With time, you could be well-versed in legal contracts, animal behavior, or the cake-baking business.
Well-versed is really just an intensified version of versed, which means "experienced in or familiar with." It comes from the obsolete word verse meaning "to turn over," so someone who is well-versed in a subject has turned it over and over again in their mind — or maybe they've read a lot of books and turned a lot of pages to study it. The word conversant, which also means "well-informed about," comes from this same root word.