1. What does it mean?
아재개그 combines 아재, a familiar form associated with 아저씨, and 개그. It labels a groan-worthy joke, often one built on a similar sound or literal interpretation.
Also written or said as ajaegaegeu
2. The nuance
The speaker does not have to be a middle-aged man. It can be affectionate teasing, and part of the fun may be how deliberately unfunny the joke is.
3. Examples
“동생이 아재개그를 하길래 웃기보다 먼저 한숨이 나왔어.”
My younger sibling told a dad joke, and I sighed before I laughed.
“제목을 이용한 아재개그라 한국어 소리를 알아야 이해돼.”
It's a pun on the title, so you need to know the Korean sounds to get it.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.