*phew* this took a while
Chezz is based off of chess. Each player takes a turns to move pieces around the board and attack. The aim is to 'kill' all of the other player's pieces. The amount of pieces you've killed is represented by the graveyard above/below your profile picture.
The pieces move in a similar manner to the king in chess, where you can move one square in any direction. After moving, you can attack once. The knight pieces (the ones with swords) attack all eight adjacent squares. The archer pieces (the ones with bows) can shoot once in one direction (this is shown in the animation). If a player attacks his/her own piece, the attacked piece will not be killed.
The walls do exactly what you think they would (things can't go through them).
The profile pictures only use those colours to keep a low colour count, so you could have as many colours as you want. They just represent the players, they're not really put of the game. The players could choose any picture they want.
The frames around the pics, however, indicate the colour of your pieces.