Whats app stores the conversation in the database, that is in the phonememory(Only rooted phones can access that).
Here is the structure of the latest whats app Database:
- messages
- sqlite_sequence
- chat_list
However, it takes a backup at 4am everyday and saves it as a db.crypt# file in the sd card. There are ways to access that though, here is one : https://stackoverflow.com/a/23380177/2219600