Away goals rule is often used in cups competitions and knock-out stages competition. It is applicable for two-legged ties, i.e. when the teams play one game home and one game away. If the two teams scored the same number of goals, the team which scored more goals in the away game.
As the Wikipedia article on this rule mentions, there are some variations to this rule. For example, in semifinals of Football League Cup, this rule is only applied after the extra time. (Here is link to the current revision of the Wikipedia article.)
As far as I can say, nowadays this type of tie-breaker is used quite frequently.
Are there some major completions with two-legged ties (cups or knockout phases of tournaments) where away goals rule is not used as a tie-breaker?