Although it's a little tricky to pronounce initially, this is actually one of the first Dutch
words that I learned when I went to live in Holland
. My Dutch friends used it most often in a sense that best translates as cozy
While cozy seemed to me initially to be an old-fashioned and somewhat weak word, I eventually started using it in everyday conversations in English, since the Dutch connotations of gezellig have no other single-word equivalent in English which goes any way towards incorporating them all. I would characterize those connotations as comfortable, safe-feeling, relaxed, happy and cheerful (although the beautiful word festelijk is more often used for that), and both physically and emotionally warm. It can be applied to places and atmospheres, and occasionally to people.
I think it's a lovely word, and it's one of those which made me start wishing that there was a language which comprised those largely untranslatable words from all the languages of the world which express unique concepts or encompass meanings in unique ways... sort of a conceptual meta-language that could be used alongside existing languages. This already happens to a small degree in most languages; consider words such as schadenfreude and chutzpah, which describe concepts which everyone understands but which no single word exists for in most languages.
George Orwell, in his novel 1984, suggests that limiting the vocabulary of a language effectively limits the thought processes of the people using that language. By and large, I agree with that assessment, and it seems to me that, while technical language develops in complexity, the everyday language of humanity becomes increasingly impoverished. The more words we have for complex concepts, the more complex concepts we are likely to be able to perceive. Bring on the meta-language!