The game is played with standard go equipment (though 13x13 boards are often preferred to 19x19) and all the rules of Go apply, except that players can't pass and the pie rule is used to determine who goes first.
The goal of the game is as in Hex: to construct a connected group extending from one side of the board to the other, e.g., one side must connect the top and bottom with this stones and the other must connect the left and right sides with hers. A player can also win if his opponent has no legal moves (since suicidal moves are illegal, as in Go).
Obviously, games between skilled players will end up deadlocked. But since players can't pass, they eventually must start filling in the internal liberties of their groups. The player who has made the smallest amount of territory will presumably get killed first, so building more territory than your opponent is a secondary (but extremely common) means of forcing a connection and the win.