Other than castling, the only "strange" move that chess pieces can ever make is en passant. It seems a little odd to me that someone thought that such a "different" move was so important to include in the legal moves. When and why did this come to be?
Allowing the en passant capture is one of the last major rule changes in European chess that occurred between 1200 and 1600, together with the introduction of the two-square first move for pawns, castling, and the unlimited range for queens and bishops (Davidson 1949:14,16,57). Spanish master Ruy López de Segura gives the rule in his 1561 book Libro de la invencion liberal y arte del juego del axedrez (Golombek 1977:108). In most places the en passant rule was adopted as soon as the rule allowing the pawn to move two squares on its first move, but it was not universally accepted until the Italian rules were changed in 1880 (Hooper & Whyld 1992:124–25).
The motivation for en passant was to prevent the newly added two-square first move for pawns from allowing a pawn to evade capture by an enemy pawn. Specifically, the rule allows a pawn on a player's fifth rank the opportunity to capture the opponent's pawn on an adjacent file that advances two squares on its first move as though it had advanced only one square (Davidson 1949:16). Asian chess variants, because of their separation from European chess prior to that period, do not feature any of these moves.
The answers currently here don't fully explain the motivation for the rule. One answer quoted this Wikipedia passage:
The motivation for en passant was to prevent the newly added two-square first move for pawns from allowing a pawn to evade capture by an enemy pawn.
That is true, but it doesn't quite get to the heart of the matter: passed pawns are extremely powerful. A passed pawn is a pawn that can never be obstructed or captured by an opposing pawn. All else being equal, if one side has a passed pawn and the other doesn't, that side will usually win.
Consider this pawn structure:
8/p7/1p4p1/1PppPp2/8/2PP1P2/8/8 w - - 0 1
White has a passed pawn on the e-file because there are no black pawns that can stop it from reaching the eighth rank and queening. Black will have to rely on his other pieces (not pictured) to stop the pawn from queening, which means those pieces won't be available to do anything else.
Black's g-pawn, on the other hand, is not a passed pawn because white's f-pawn will eventually capture it if it tries to queen. That means white's pieces won't be tied down trying to stop the g-pawn.
Now look at black's a-pawn. It's not a passed pawn because it has to get past white's b-pawn before it can queen. However, if the a-pawn were allowed to advance two squares, and the en passant rule didn't exist, it would be a passed pawn because black could move it to a5 without being captured. But, since the en passant rule does exist, attempting this would be a horrendous mistake because white would capture the pawn en passant and suddenly white would have a passed pawn on the a-file.
So, in essence, the en passant rule allows you to advance your pawn to the fifth rank without fear that your opponent will simply bypass your pawn by moving two squares.
In a "real" war, if one army moves a unit adjacent to the enemy army, it is likely to provoke a return "fire" from the enemy army. In chess, this "fire" is expressed by capturing.
En passant is a compensation for a pawn being allowed to advance two spaces on its first move. A unit that marched at "double time" would probably rest after half the journey, then resume its march to the destination.
So assume that the pawn marches one square, rests, then tries to complete its march to the second square. But if there is an enemy pawn on the adjacent file, that pawn might start a "duel" with the marching pawn when it is resting. That would result in an en passant capture.