In addition to standard rules, all captures causes an "explosion" of which all surrounding pieces (Not pawns!) on the adjacent squares (one square horizontal to, vertical to, and diagonally from the capture) and the capturing piece is removed from the board. Therefore, it is illegal to capture a piece that results in your king blowing up, nor can a king capture any piece (both of these actions would end his life).
A special note for Atomic games is that kings can be connected by moving one into the adjacent squares of the other sided king (and in most cases leads to a draw). When the kings are connected, checks do not apply. As it is illegal for a capture to blow up your own king, it is not possible to capture the other king directly. To "win" such an endgame, it would be necessary to cause the kings to detach by zugzwang, or that a piece of the opposite color is exploded while the king is around it.
A checkmate is declared if a king is put in check, even if the king isn't immediately blown up, given that the king has no escape squares.
Any move that results in blowing up the opposite King will claim an immediate victory, overriding all checks.
Knights, Rooks, Queens, Bishops and Pawns also develop their respective tactics in Atomic chess. They preserve their material value from the Piece Relative Values in Standard Chess.