The day after tomorrow (two days after today). Commonly spoken as "asatte."
明後日、友達と映画を見に行きます。
I'm going to the movies with a friend the day after tomorrow.
ビザの申請期限は明後日ですから、忘れずに出してください。
The visa application deadline is the day after tomorrow, so don't forget to submit it.
明後日（みょうごにち）は出張で東京に行きます。
I will go to Tokyo on a business trip the day after tomorrow (myōgonichi).
Used colloquially to say that something (a direction, remark, idea) is completely off or mistaken — e.g. "looking the wrong way."
彼の説明はいつもあさっての方向を向いていて、要点が分からない。
His explanations are always off in the wrong direction, so I can't understand the point.
そんな意見はあさっての方向だ。もっと現実を見た方がいい。
That opinion is way off. You should be more realistic.