Only the two of them; being alone together (no one else present).
毎晩、彼と二人っきりで話す時間が楽しみだ。
I look forward to the time each night when it's just the two of us talking.
親に知られないように、二人っきりで計画を立てた。
We made plans alone together so our parents wouldn't find out.
その部屋には二人っきりしかいなかった。
There were only the two of them in that room.
デートの最後に二人っきりになって告白した。
At the end of the date, we ended up alone together and I confessed.