A full rook means winning a rook without any compensation. For example, you might win a rook in exchange for a pawn, which wouldn't be a "full rook": it's a rook minus a pawn (4 points in estimated material). But a "full piece" means you win the piece without getting even a pawn for it. Another situation is when you win a rook in exchange for a knight or bishop, which is usually known as "winning the exchange".
In chess literature, I've often come across the phrase in the title, so it made me wonder:
How do you win half a rook?