A flight of steps is a continuous run of steps between landings. So if you go up 10 steps, come to a straight flat area for awhile, and then climb another 10 steps, you have gone up 2 flights of steps. Inside a building the landing is often the area where the stairs change directions. So you go up one flight of steps in one direction, turn at the landing, and go up another flight of steps in the opposite direction. Did that make sense?
A flight of stairs. A single flight of steps or stair would be a single block of stairs (usually less than 20), large runs of steps will often be divided into multiple flights with large flat areas in between in order to match the desired incline.