A pawn on e2 has two moves, e3 and e4. Once it stands on e3 it has one, e4. That is the whole of pawn movement, and it is the simplest rule in chess until you notice that the pawn is the only piece that does not capture the way it moves.
h 1 Straight ahead to move, diagonally forward to take, one square at a time, and only ever towards the far end of the board. Never sideways. Never back. The full rules for taking are on how pawns capture, because that is where most of the confusion lives.
From its starting rank a pawn may go two squares instead of one. White’s pawns start on the second rank, Black’s on the seventh, so 1.e4 and 1…d5 are both double steps.
The condition is that the pawn has not moved yet, and both squares have to be empty: a pawn cannot jump, so a knight on e3 stops a pawn on e2 from reaching e4 as surely as it stops it reaching e3. And the right does not expire. A pawn still sitting on h2 on move sixty can play h4 if it wants. More on the two-square move, including the trap it creates.
The part beginners miss about pushing pawns at all is this: the two squares a pawn attacks travel with it. A pawn on e2 covers d3 and f3, squares nobody is fighting over. The same pawn on e4 covers d5 and f5, right in the centre. That is what 1.e4 actually buys, and it is why the centre pawns are the ones that get pushed first.
A pawn in front of a pawn is a wall
This is the position new players stare at with a hand hovering over the board.
h 1 Nothing at all can happen between these two. The white pawn cannot take straight ahead, and it cannot go round. This is why a blocked centre changes the whole character of a game: with the pawns locked, the play goes to the wings and the knights become better than the bishops.
Count them, while you are looking. Sixteen pawns start the game, half of everything on the board. Most people treat them as the stuff standing in front of the real army. They are the terrain, and the pieces have to live on whatever the pawns leave behind.
En passant, the rule people refuse to believe
The double step created a loophole, so the Laws closed it. If a pawn uses its two-square move to run past a square that an enemy pawn attacks, that enemy pawn may capture it anyway, as though it had only moved one square.
Say White has a pawn on e5 and Black plays f7-f5, sliding past e5’s reach. White may answer exf6, taking the black pawn off f5 and landing on f6. The capture is legal on that turn and never again: if White plays something else, the chance is gone for good. Two conditions, no third. The capturing piece has to be a pawn, and the reply has to be immediate. The first time I played it against a junior in a league match, he called the arbiter over. It is a real rule and there are puzzles built on it.
The far end
A pawn that reaches the eighth rank (or the first, for Black) does not stop there. It is replaced, as part of the same move, by a queen, rook, bishop or knight of its own colour. Not a king, not a pawn, and you do not have to have lost your original queen first. Three queens on the board at once is perfectly legal, and it happens in club games more often than you would think.
e That is what all the endgame fuss is about. A pawn is worth one point on move ten and worth nine on move sixty, which is why grandmasters will give up a bishop to create a passed pawn and why promotion decides more games than any tactic. Once in a while the right answer is a knight, because a knight can give a check no queen could, or even a rook, because a new queen would leave the enemy king with no legal move and draw the game on the spot.
Every push is permanent
Every other piece can undo its own move. A pawn cannot. Push it and it is pushed for ever, so the holes it leaves behind stay behind it for the rest of the game. That is the reason my first question about any pawn move is not what it attacks but what it stops defending, and it is the reason pawn structure is a subject at all. Before you play g4 in front of your own king, look at what h3 and f3 will look like when g4 can never come home again.