1. What does it mean?
In dating talk, 애프터 borrows 'after' to mean seeing someone again after a 소개팅 or first arranged date. 애프터 신청 asks for that next meeting.
Also written or said as aepeuteo
2. The nuance
It is a noun-like Korean use, not a complete English expression. An accepted 애프터 means agreeing to meet again, not automatically agreeing to an exclusive relationship.
3. Examples
“첫 만남에서 이야기가 잘 통해서 애프터를 신청했어.”
We got along well at the first meeting, so I asked for a follow-up date.
“애프터는 거절했지만 시간을 내 줘서 고맙다고 전했어.”
I declined a second date but thanked them for making time.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.