Mahjong (or Mah-jongg) is a tile-matching game that originated in China over a century ago. Today it is played all over the world, and like many other traditional games, there are now a multitude of free online variations of Mahjong you can play in a Web browser. The rules are very simple; you select matching tiles from the vertical edges of the layout to remove them, until all the tiles are gone. These are some of the better free Mahjong games I've come across so far.
This free Mahjong game at Candystand has a nice selection of backgrounds and layouts to choose from. Your score is based on the time it takes you to clear the puzzle and bonuses for matching 4 like tiles. If you get stuck you can shuffle the remaining tiles, but it will cost you some points. Registered users can submit high scores to the site.
Miniclip's Mahjong Daily is entirely free and features a new layout every day. It uses only one tileset and sticks with oriental-themed music, but it's a good place to get a free daily fix of tile matching.
Although there is still tile matching involved, this game from Pogo is actually a major departure from traditional Mahjong. Mahjong Safari has only one layer of tiles, and the idea is to connect matches with a line that has no more than 2 corners. It's faster-paced and more arcade-like than regular Mahjong, so it may appeal to those who find standard Mahjong a bit dull.
A fairly traditional Mahjong game, Cafe.com's Temple of Mahjong 2 has several different themes and tilesets to choose from. It always uses the same layout shape, but your performance is timed. Rooms are available where you can compete with other players for the best time. If you join Cafe.com you can purchase "boosts" through their microtransaction system.
Mahjong Tower has plenty of layouts to choose from, and there is also a free layout editor it you want to create layouts of your own, although it may take a little fiddling if you don't understand Chinese. About the only thing missing is the soothing music that most players are accustomed to hearing during their Mahjong sessions.