Rook, knight, bishop, queen, king, pawn. Six names, and two of them get mangled every day of the week, because the piece with the battlements is not a castle and the one shaped like a horse is not a horse.
You get sixteen of them a side, so thirty-two on the board: eight pawns, two rooks, two knights, two bishops, one queen and one king. That is the whole inventory, and if a set from a charity shop is short of something, count the pawns first. They are the ones that go missing.
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e h 1 Where the two wrong names come from
Castling is a move, not a piece, so calling the rook a castle leaves you saying that you castled your castle. The real name comes from the Persian rukh, a chariot, which suits a piece that charges the length of an open file.
The horse is a knight, and that one is pure appearance: the carving is a horse’s head, the piece is the rider. Say horse at a club and somebody will correct you within the hour.
The other four names nobody argues about: king, queen, bishop, pawn. Notation gives five of them a capital letter, K, Q, R, B and N, with the knight taking N because the king already has K. Pawns get no letter, so e4 is a pawn move and Ne4 is a knight move.
There is also a split inside the word “piece”. Bishops and knights are the minor pieces, rooks and queens the major ones, and players use “piece” in that narrow sense when they say they have won one. That distinction has its own page on minor and major pieces.
One line on what each of them does
The pawn goes forward one square, or two from its starting rank, and captures diagonally, so a pawn on e4 takes on d5 or f5 and never straight ahead.
The rook runs any distance along a rank or a file, and is worth five pawns.
The bishop runs any distance on diagonals and stays on one colour for the entire game. The c1 bishop dies a dark-squared bishop.
The knight jumps in an L, two squares then one across, and is the only man allowed to leap over anything in the way. From f3 it reaches eight squares; from h1, two.
The queen combines rook and bishop, up to twenty-seven squares at once from the centre, and is worth about nine pawns.
The king steps one square in any direction and may never move onto a square an enemy piece covers. It has no point value because losing it loses the game.
Learn the names first, then the values, because the values are what stop you swapping a rook for a knight and wondering why the position went sour: what each piece is worth. And the knight, the one with the silly name, is the piece that wins most beginner games, usually by forking a king and a queen from a square nobody was watching.