Put a white king alone on e4 and count his squares: d3, e3, f3, d4, f4, d5, e5, f5. Eight. Move that same king to h1 and the count drops to three (g1, g2, h2). One step at a time, in any direction, diagonals included. That is the entire movement of the most important piece on the board, and it never improves.
He captures exactly the way he moves. Put a black pawn on d5 in that picture and the king simply takes it and stands there. There is no special capture rule for the king, and no type of piece he is forbidden to take, apart from the enemy king, which he can never get close enough to touch. That one has a page of its own.
The rule that shapes everything: he cannot walk into check
The FIDE Laws give the king two ways of moving (Art. 3.8): to an adjoining square that is not attacked by an enemy piece, or castling. Read the first half twice. Not attacked. A square can be empty, inviting and completely illegal, because a pawn covers it, or a bishop three squares away covers it, or the enemy king covers it.
So a king almost never really has eight squares.
That rook has not given a single check and has already taken a third of the king’s freedom. It is the whole idea behind king and rook against king: you never chase the king around, you build a wall he cannot cross and then shrink the box.
Two consequences that catch new players out. First, a king may not capture a defended piece, because the square he lands on is attacked. Second, a king may not run away from a check along the checking line: with a black rook on e8 giving check to a white king on e4, the move Ke3 is illegal, since e3 is still on the e-file. You have to step off the line, to d3, d4, d5, f3, f4 or f5, or block the check with another piece, or capture the rook with one. Those are the only three ways out of check there have ever been.
And if there is no such move at all? Then it is checkmate if he is in check, and stalemate, a draw, if he is not. One rule, two very different results, and beginners throw away won games on the difference every day.
Castling, the one time he moves two squares
Once per game the king travels two squares towards a rook, and that rook jumps over him and lands on the square he crossed. It counts as one move.
8 
7 
2 
a c 
d e h 1 There are five conditions, and most beginners can recite three of them: neither the king nor that particular rook has moved, the squares between them are empty, the king is not in check, he does not pass through an attacked square, and he does not land on one. Notice what is absent from the list. The rook may be attacked, and the rook may cross an attacked square, and you can still castle. Only the king is fussy. The rest of the small print sits on what castling is and when you cannot castle.
He is a liability, then suddenly he is a piece
For the first twenty moves the king wants to be hidden. Castle early, leave the three pawns in front of him alone, and when you have a quiet move to spare give him an escape square. That little hole has a name, luft, and it is the cheapest insurance there is against back-rank mate.
Then the queens come off and the same slow piece becomes one of your best. A king on e5 attacks eight squares, defends his own pawns for nothing, and cannot be captured by anything. In a king and pawn ending the difference between a king on e1 and a king on e4 is usually the difference between a draw and a win, which is why the first ending anyone should learn is the opposition.
He is also faster than he looks. From e1 to e8 is seven moves, and from a1 all the way to h8 is also seven, because a diagonal step costs exactly what a straight one costs. A king chasing a pawn down the board can cut the corner for free, and that is the whole basis of the rule of the square, the trick that tells you in one glance whether the pawn escapes.
Try this before you read anything else
Set up a board with two kings and nothing else, one on e1 and one on e8, and walk them towards each other. They will stop with one square between them every time, and you will feel the rule from the first diagram in your hands instead of reading it. Now add a single white pawn on e2 and try to promote it against your own best defence. Six or seven attempts and you will have worked out the opposition on your own, which is exactly how I teach it to eight-year-olds.