This is Usurp, a simple two-player checkers-style game played on a 3-dimensional 4x4x4 grid (which is represented as four 4x4 grids placed horizontally across the screen: the leftmost is the "top", the rightmost is the "bottom").
The object is to rid the board of all of your opponent's pieces. You may move in any one of 26 directions one step at a time: up, down, left, right, diagonally and all of those directions PLUS up and down levels (or JUST up and down levels). If you move your piece onto a grid square occupied by your opponent, your opponent's piece is removed. If there is another one of your opponent's pieces on one of the grids in the EXACT same direction as your piece just went, you will remove that piece as well (similar to multiple jumps in checkers).
At the title screen, pressing [HELP] gives you more detailed instructions but this should give you an idea of how to play if you want to skip those screens. The game requires joysticks and pieces are moved by selecting them, pressing [FIRE], selecting a new position and pressing [FIRE] again. Move between levels with the [TAB] key.