Looking at alternatives to the bog standard starting layout we have Horde Chess, Pawns Game and Peasant's Revolt. Then moving beyond the traditional eight by eight grid of squares we have stuff like Hexagonal, Masonic, Circular or even the Tri-D boards as seen on Star Trek. As for rule variations there's stuff like Ability Absorption, Kamikaze Pieces, Three-Check and my personal favorite Take-all in which the king has no special rules associated with him and the game doesn't end until all pieces on one side are captured.
Moving on, one of the greatest criticisms of chess is its dry analytic nature. For some that's the whole point, but I've always enjoyed games more when there are some random elements thrown in. In the context of chess there's Dice Chess and No Stress Chess, where pieces get to move based on the results of dice and cards respectively. Then there are also interesting ways to randomize things with Synchronous Chess (in which moves are recorded, shown, then made simultaneously) and Viennese Chess (in which a partitioning screen is used during setup while each player arranges the desired location of their pieces in secret on respective sides of the board).
The number of players can also vary beyond the standard one on one matches normally associated with chess. Three player or even four players are possible using unconventional board layouts. Fortress Chess was popular in pre-soviet Russia and has an interesting optional rule in which it is possible to revive a check matted ally. Meanwhile three player chess games usually have rules in which the first to checkmate another wins, thus discouraging alliances almost entirely.
Lastly, things get really weird when you consider some of the chess derived games out there such as Dragonchess, Jetan, Shogi or Antichess (a game in which capture moves are mandatory and the winner is the first to loose all their pieces). Super King, Scottish Chess and Kung-fu chess are just a few of other more crazy versions of this game people have come up with over the years. The most extreme of these has to be boxing chess, but I'm getting too far off topic.