A pitcher earns a complete game if he pitches the entire game for his team regardless of how long it lasts. If the game is shortened by rain or if it lasts into extra innings, it counts as a complete game if the pitcher was the only pitcher to record an appearance for his team.
Complete games are considered valuable in baseball for a variety of reasons. First of all, if a pitcher hurls a complete game, it generally means he pitched well -- usually well enough for a victory and sometimes even well enough for a shutout (a complete game with no runs allowed). But complete games are also important because they give the bullpen an entire day of rest, keeping it fresh for the future.
Typically, the league leaders in complete games are pitchers who can keep their pitch counts low and avoid walks (and sometimes strikeouts, too, because strikeouts typically require more pitches to achieve than other types of outs). It takes a major effort for a pitcher to toss a complete game, as fatigue generally sets in during the later innings.
As bullpens have grown in importance over the past few decades, complete games have become increasingly rare.
In A Call
"goes the distance"