1There; that place over there — a location distant from both speaker and listener.
山の向こうに、あそこに小さな村が見える。
You can see a small village over there beyond the mountain.
駅はあそこです。まっすぐ行ってください。
The station is over there. Go straight ahead, please.
あそこへ行ったことがありますか？
Have you ever been there?
子どもたちがあそこで遊んでいる。
The children are playing over there.
2A euphemistic, colloquial term for the genitals (often used with, or for, children).
小さい子には『あそこは触ってはいけない』と教える必要がある。
You need to teach small children, "You mustn't touch down there."
医者に『あそこに痛みがありますか？』と聞かれた。
The doctor asked, "Do you have any pain down there?"
3That far; to that degree; that point — a degree that is psychologically or abstractly distant from both speaker and listener.
彼女があそこまで怒るとは思わなかった。
I didn't think she would get that angry.
試合はあそこまで接戦になった。
The match ended up that closely fought.