How should I model a database structure to store word information on flashcards?
Generally speaking, databases don't store arrays. 

It sounds like you need several tables to track different things.


A "users" table which will store the userID (primary key), user's name, and so on.
A "base_word" table to store the base word, a BaseWordID (primary key) and such.
A "Users_Words" linking table. This table would have it's own primary key, a UserID foreign key to link to the Users table, a BaseWordID foreign key link to the base_word table, and a Learnt flag to track learnt or not (possibly other columns, too?)


If they need access to the data "offline," that's not duplicating the data. That's generating a "report" from the data. This might be an HTML page derived from the data that is printed (or printed to PDF), for example. Or an Excel file dump of the data.