Put a knight on e4 and count its moves: c3, c5, d2, d6, f2, f6, g3, g5. Eight. Now put the same knight on a1 and count again: b3 and c2. Two. Nothing has changed about the piece. It just moved to a place where most of its moves fall off the edge of the board.
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e That ratio is the entire case for controlling the centre, and it applies to every piece except the rook. A bishop on d4 sees thirteen squares, on h1 it sees seven. A queen in the middle covers twenty-seven, in the corner twenty-one.
The four squares, and the sixteen around them
The centre is d4, e4, d5 and e5. When somebody says a game was decided by the fight for the centre, those four squares are what they mean.
The extended centre is the 4x4 block from c3 to f6: c3 to f3, c4 to f4, c5 to f5, c6 to f6. Knights on c3 and f3 sit in it, and so does the bishop on c4 that every beginner plays, which is why those moves feel right long before anyone explains why.
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d e f g h 1 Control does not mean standing there
This is where beginners go wrong. Control means attacking the square, not occupying it. A pawn on e4 occupies e4 and attacks d5 and f5. A knight on f3 attacks e5 and d4 from a square that is not central at all. A bishop fianchettoed on g2 looks straight down the long diagonal at d5 from the edge of the board, which is why the fianchetto is a central move dressed up as a flank one.
Occupying without controlling is the classic mistake. Push a pawn to e5 on move three, and if you cannot defend it, your opponent hits it with d6, Nc6 and Bf5 and you spend the game babysitting a pawn. Occupation is only as good as the support behind it. The related idea, how much room your pieces have behind their pawns, has a page of its own on space.
What this means for your first ten moves
Start with a centre pawn. 1.e4 or 1.d4 takes a central square and opens lines for two pieces at once; there is more on the best first move if you want the argument in full.
Then bring the knights to f3 and c3 (or f6 and c6 as Black), where each one attacks two central squares from safety, rather than to h3 and a3 where they attack one and get in the way. Then castle, then put a rook on the file where the pawns are about to be traded.
That is four ideas and it will get you through the first phase of any game. They are all in the opening principles, and if you want to see what happens when one side wins the fight for the middle, the opening puzzles are all positions where somebody has already lost it.