In crossword terminology, a "seat" is simply a synonym for a numbered grid square or light—the individual white cell where a single letter is written.
Constructors use "seat" metaphorically to indicate a location or position. For example:
Clue: "Seat of government" (5 letters)
Answer: CAPITAL
Here, "seat" means "location/center," not a physical chair. The answer goes in the numbered squares (e.g., 1-Across).
Key Takeaway: If a clue says "Seat of X," think "Location/Capital of X." If the puzzle instructions mention "seats," they just mean the numbered boxes in the grid.