The Away Goals Rule is a simple idea in theory, that gets confusing in practice. Normally, it only applies in a home-and-home series like the knockout stages of the UEFA Champions League and is used as a series tie-breaker. Away goals are goals scored away from your home stadium. If a series is tied on aggregate score (the combined score from both games) at the end of the second-leg, the team that scored the most away goals wins.
The English League only uses the Away Goals Rule in the League Cup, and then it only counts after an overtime period has been played.
Remember, away goals aren't just for the first leg of a two-game series. The away goals scored in the second leg also count in determining a winner.
Using the second leg of the Chelsea - Liverpool Champions League quarterfinal series as an example: Chelsea leads 3-1 and all three of their goals were scored away. For Liverpool to force an overtime period, they would need to win 3-1. To win outright, they would need to beat Chelsea by three goals.