There are three pieces known as pawns, the two eastern pawns, and the western pawn. They all move a single step orthogonally forward, but differ in how they capture and other small details.
The western pawn is the most complicated variety. It can only move directly forward, but can only capture diagonally forward.
The western pawn promotes upon reaching the final rank. Originally this was to the ferz and was indicated by turning the pawn upside down.
Later games have more variety. One variation is to have the pawn promote to the piece that occupied the promotion square at the start of the game. What happens on the king's square varied.
In Italy, when the queen replaced the ferz, pawns promoted to queens and promotion became a major endgame threat. Since the queen is the most powerful, it was conceded that it was acceptable for pawns to promote to lesser pieces as well, so the option of "under promotion" to any piece was added.
To speed up the opening, the Italian game allows a pawn on it's initial move to go two spaces. This is a slide, not a jump, so the intervening space must be empty.
The rule was intended to speed the game up without altering strategy. However, the double move allowed a pawn to "pass" an opposing pawn by skipping over a square it threatened. To correct this deficiency, the en passant ("in passing") rule was added.
On the turn immediately following a double pawn push, any pawn attacking the passed over square may move to that square and take the pawn that made the double move.
The author considers it a bug that only pawns can take advantage of this rule. Pieces can be passed with impunity.
Xiangqi Pawn (卒/兵)
The xingqi pawn is fairly simple. They move and attack directly forward. When they cross the midpoint of the board (the river), they gain the ability to move (and attack) to each side as well. As they never gain the ability to move backwards, no indication of promotion is required.
Shogi Pawn (歩兵)
In shogi, the pawns are quite simple. They move and attack directly forward. They promote to move as a gold general. Unlike other games, promotion is not unique to pawns as nearly all shogi pieces promote.
In shogi variants with drops, pawn drops are subject to two special restrictions. Firstly, only one (unpromoted) pawn (per side) can be in a file. Secondly, a pawn may not be dropped such that it gives checkmate.