The edge of a hex board is just what you think it is -- the board is a rhombus with four edges. Each player "owns" two opposite edges of the board in game of hex, and their objective is to connect their two edges with a path formed by their own pieces.
For example, in this game here
red has managed to create a (rather twisty) path from the top edge to the bottom edge, and hence has won the game. Blue's goal was to create a path from the left edge to the right edge, however blue can no longer do this because red's path blocks all blue paths.