In a penalty shootout, each team alternates for five sets of penalty kicks. After those five sets, whoever has the most penalty kick goals wins the game. If a penalty shootout is still tied after the initial five sets, then the shootout goes into sudden death.
Penalty shootouts are used to decide the winner in a cup match when the two sides have finished the game with the same number of goals - ie a tie. In a league match, the result just goes down as a draw and each team gets (usually) one point, whereas a team that wins the game would get three and the losers zero.