1. What does it mean?
짱나 compresses 짜증 나 in speech-like texting. It vents irritation about a delay, a nuisance, or something that has gone wrong; 짱나다 may also appear as a dictionary-style slang form.
Also written or said as 짜증나, jjangna
2. The nuance
It is not related in meaning to 짱, 'the best.' The expression is blunt and casual, and directing it at a person can hurt even when you intend only to vent.
3. Examples
“양말 한 짝만 자꾸 없어져서 짱나.”
It's so annoying that just one sock keeps disappearing.
“버스 놓친 건 짱나지만 다음 차는 금방 오네.”
Missing the bus is annoying, but the next one's coming soon.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.