Churuya is an original character created by the artist Eretto, based on the design for Tsuruya from the novel/anime/manga Suzumiya Haruhi no Yuuutsu. Churuya is depicted as a super-deformed version of Tsuruya. The two have fairly dissimilar personalities. Whereas Tsuruya is easily excitable and laughs a lot, Churuya seems more simple-minded, and has a deep appreciation for smoked cheese. Her signature catchphrase is nyoro~n.
Churuya first appeared on Eretto's website in a now famous 4koma (post #47838). Churuya inherited Eretto's love of smoked cheese. Notably, in the last panel Churuya utters her catchphrase nyoro~n when she becomes disappointed by the lack of smoked cheese. Likely one major reason for Churuya's popularity is the sense of schadenfreude at Churuya's misfortune. The regularity of her misfortune can be likened to that of Hans Moleman from The Simpsons.
From the very start, fan reaction to Eretto's Churuya comics was wildly positive. After being linked on 2ch.net, Eretto's site would frequently go down because of all the traffic. Eretto continued to write new Churuya-san 4koma, and has even anthologized them in doujinshi.
Churuya has become so popular that other artists have begun to parody it. Usually the one common factor is the last panel, where a character drawn in the style of Churuya (complete with :3) says nyoro~n or some variation thereof.