Demi-chess was invented in 1986 by Peter Krystufek. It was designed to be a game for beginners. The game is actually quite nice to play with young children: while it isn't as rich as the real chess, it helps to learn basic principals like watching out that your pieces cannot be taken. I played the game against my (then) six-year old daughter and it was lots of fun.
The game is played on a board with eight rows and four columns. The opening setup is as follows:
King a1; Rook d1; Knight c1; Bishop b1; Pawn a2, b2, c2, d2.
King a8; Rook d8; Knight c8; Bishop b8; Pawn a7, b7, c7, d7.
Castling is as to be expected. Pawns can also promote to queens. All rules are further as in orthodox chess.
The game was described in Pritchard's The Encyclopedia of Chess Variants.