Imagine the board is an infinite grid. Other rules are the same as conventional chess.
Note that this would have two alternatives with regards to pawn promotion: either they can't be, and just continue indefinitely in their direction, or you allow them to promote at the 8th (or some other) rank as normal.
It might be hard to mate in this variant: most endgames will be a draw. For instance, an endgame with a queen and a king against a lone king is a draw.
I thought of a rule to prevent a lot of the problems which occur with an infinite board.
It is the rule of "withdrawal from battle". If a piece moves, or if other pieces move away from it, so that there are eight or more empty spaces between that piece and any other piece, it is considered to have withdrawn from battle and is removed from the board.
Using this rule, a Q+K vs K is a win. Use the Queen to hold off the other king while your king moves eight spaces away (so there are seven intervening empty spaces), then move the queen to the other side of your king. The opponent's king is then vanquished by having eight empty spaces between it and any other piece.
Concerning pawn promotion, I would allow them to promote if they succeeded in getting past all of the opponent's pieces.