Kyun (きゅん) is the little squeeze in your heart when something is very sweet, cute or romantic.
In a film, the shy boy finally gives her his umbrella and runs off into the rain. Your chest goes tight for a second. That is きゅん.
What it means
きゅん is short and sudden: one moment that touches your heart. It can be romance, but also a sleeping puppy, a kind word or an old song. It is always a soft, happy feeling.
How to use it
Use it with と します: きゅんと します, きゅんと しました. For a feeling that keeps going, double it: きゅんきゅん します. A scene that does this is a むねキュン scene (むね is your chest).
みて！ こいぬの しゃしんです！
Look! A photo of a puppy!
…きゅん。
...Aww, my heart.
ふじさんも こいぬが すきですね。
So you like puppies too, Fuji.
Close cousins
どきどき is your heart racing, from excitement or nerves, and it goes on and on. きゅん is shorter and sweeter: one squeeze. ときめく is the verb for that fluttering feeling, and its noun ときめき is a favourite of song titles.
In manga and anime
In romance manga, a heroine gets small hearts and キュン beside her face when the boy smiles. Fans write キュン or むねキュン about their favourite scenes.
Can you use it?
Yes, with friends, about films, pets and anything sweet. It sounds young and cute. You would not use it in a meeting.