The phrase means returning to the starting point, typically after a plan has failed or progress has been reversed. A job applicant rejected at the final stage is back to square one. A business that loses its main client may find itself back to square one. The implication is disappointment — work done has been undone.
The most widely cited origin connects to BBC radio football broadcasts in the 1920s and 1930s. To help listeners follow commentary without being able to see the pitch, the Radio Times printed a numbered grid of the football field divided into eight squares. Square one was the goalkeeper's area. When the ball returned to the goalkeeper after an attacking move broke down, the commentator would note that play was back to square one — at the start, with nothing gained.
Some researchers question the BBC theory, noting that the grid system may not have been as formally standardised as later accounts suggest. Other candidates include the board game Snakes and Ladders, where landing on the wrong square could send you back to the start, and Hopscotch, where a player returning to square one begins again. The phrase is firmly established in English from the 1950s regardless of which origin is correct.
It is primarily British in origin, traceable to BBC radio and later British print media of the 1950s, though it is now used across all varieties of English.
'Back to the drawing board' carries much the same meaning — starting again from scratch after a failed attempt. 'Start from scratch' is another equivalent.