Hex is a board game played on a Hexagonal grid, normally parallelogram-shaped (or rhombus).
Players have two colors, say red and black. They alternate turns, filling in a hex with their color. Red's goal is to form a red path connecting the top and bottom, and Black's goal is to form a patch connecting the left and right. The game can never end in a tie, because the only way to prevent your opponent from forming a path is by forming a path yourself.
When the sides of the grid are equal, the game favors the first player, by a standard strategy-stealing argument. (If the second player has a winning strategy, the first player could steal it by making an irrelevant move and then follow the second player's strategy. The extra move can only help the first player, i.e. if the strategy ever called for moving on the square already chosen, the first player makes another random move.)
There are two ways to make the game fairer. One way is to make the second player's sides closer together, playing on a parallelogram rather than a rhombus. Another way is to allow the second player to choose his color after the first player makes the first move, which encourages the first player to make a weak initial move.
Hex had a brief incarnation as the game show Blockbuster.