Yōkaï no Mori contains two simpler versions of Shogi to introduce the game to new players and youngsters.

In the classic version, each player has only four pieces – kodama, tanuki, kitsune, and koropokkuru – and they're trying to either catch the opponent's koropokkuru or move their own koropokkuru to the opposite side of the game board. The board is just 3×4 squares, so the game is rather short. As for how to play:

The tanuki moves one space orthogonally.
The kitsune moves one space diagonally.
The kodama is directly analogous to the pawn/soldier of Shōgi and can be promoted to a kodama samurai, equivalent to the gold general in Shogi (moving one space orthogonally or one space diagonally forward).
The koropokkuru is analogous to a king in Chess, except that he can win through promotion as long as it doesn't leave him in check.
Even on this 3×4 board, you are still able to drop in pieces that you captured from your opponent!
Unlike in Shōgi, you can drop a kodama for mate or have two kodama in a column.

The expert version has a larger 5×6 game board with each player having eight pieces (3 kodama, 2 oni, 2 kirin and 1 koropokkuru).

The kirin moves equivalent to the gold general in Shogi.
The oni moves equivalent to the silver general in Shogi (moving one space diagonally or one space straight forward) and can be promoted to a super oni, equivalent to the gold general in Shogi.

The game also adds more rules from the full Shōgi game, in particular:

You win only by capturing your opponent's koropokkuru, not by reaching the opposite side of the game board.
You can't have two of your own kodama in the same column/file.
You can't drop a kodama for mate.

The Japan Shogi Association created the rules for this game in order for players to learn important concepts in Shōgi, such as how to handle the generals effectively, why pawn promotion is so important, and how to mate the opponent's king with successive piece drops.