In chess, a king walk is the phrase used to describe a long journey across the board taken by the king, generally because it is being hunted by enemy forces. A king walk will often end in checkmate, but not always: there are many examples of a king walk ending in the safe ensconcement of that piece in a totally different area of the board.
Often a player will sacrifice material in order to extract the enemy king from its hideout and force it across the board, because an exposed king causes immense problems for a player attempting to defend. The king can only move one square in any direction on each move, and therefore on its own is a very weak piece in the opening and middle game, only becoming strong in the endgame, when there are too few pieces on the board to form a mating net.
One of the most famous king walks in chess history is in the game Edward Lasker - Sir George Thomas, played in London in 1912:
1.d4 f5
2.e4 fxe4
3.Nc3 Nf6
4.Bg5 e6
5.Nxe4 Be7
6.Bxf6 Bxf6
7.Nf3 O-O
8.Bd3 b6
9.Ne5 Bb7
10.Qh5 Qe7 (Now follows the king walk, in which White sacrifices his queen, and Black's king is forced to traverse the entire board, ending up checkmated on White's first rank.)
11.Qxh7+!! Kxh7
12.Nxf6+ Kh6
13.Neg4+ Kg5
14.h4+ Kf4
15.g3+ Kf3
16.Be2+ Kg2
17.Rh2+ Kg1
18.Kd2++
The game can be viewed with diagrams at Chess Corner: http://216.118.77.254/games/fascinating/int1.htm