The story
Gezellig is the Dutch word visitors hear first and understand last. It comes from gezel, 'companion', a word that in the Middle Ages also meant a journeyman in a craft guild, so at its root it describes the pleasure of being in company.
That root still decides how it is used. A room can be gezellig, mostly because it looks like a good place to spend time with people, with low lamps and a big table. A person is gezellig when they are easy company. An evening can be gezellig, and so can a queue at the bakery that turns into a chat. The opposite, ongezellig, is a real reproach: leaving a party early is ongezellig, and so is staring at your phone over dinner.
English usually reaches for cosy, but Dutch already has a word for cosy, knus, and it is about the room. Gezellig is about how people feel together, which is why translators often need two English words, cosy and convivial, and still lose something on the way.
A scene: a brown café in Utrecht on a wet Tuesday in November. Four colleagues meant to have one beer after work. Two hours later the candle on the table has burned down, someone has ordered bitterballen, and nobody has mentioned going home. Ask any of them how it was and the answer is one word: gezellig.
The word travelled with the Dutch. Afrikaans has gesellig, and Papiamentu borrowed gezellig itself.