In sports, 'away' refers to a match played at the opponent's venue (away game) or the opponent's side/goal.
今日はアウェイでの試合だから、応援は少し厳しいかもしれない。
Since today's match is away, the cheering might be a bit sparse.
アウェイゴールのルールが勝敗を左右した。
The away-goal rule decided the outcome.
チームはアウェイで強い戦い方を見せた。
The team showed a strong performance on the road.
A feeling of being out of place or uncomfortable in unfamiliar surroundings; colloquially described as feeling 'away'.
新しい職場では最初、アウェイな気分だったが次第に慣れた。
At the new workplace I felt out of place at first, but gradually got used to it.
留学先では文化の違いでアウェーを感じることがある。
While studying abroad you may sometimes feel out of place due to cultural differences.
観客が全員相手チームのサポーターで、スタジアムは完全にアウェイだった。
With the stands full of the opponent's supporters, the stadium felt completely like away territory.