A crossword was a word puzzle.
The Third Doctor worked on a crossword while Liz Shaw examined rock samples which purportedly came from Venus. (COMIC: Doctor Who and the Rocks from Venus) He was later thrown out of the Diogenes Club when his seventh incarnation showed him the answer to the crossword he was doing, causing the Doctor's earlier incarnation to shout at his older self and break the rules of silence. (PROSE: All-Consuming Fire)
According to the Seventh Doctor, there was no point doing a crossword puzzle if one kept their thumb at the back of the book to check for answers. Ace figured that if the Doctor wanted to do a crossword, he would go back in time, "nobble" the crossword setter, and then write it himself. (AUDIO: The High Price of Parking)