Little square brown creature with a gaping toothy mouth, born in Japan. Big in Japan, too. For some reason used as a mascot (not, presumably, by the artists who made the domokun) against the perfectly natural, pleasant and healthy practice of masturbation. See also Religion.
Domo-kun is a small brown open-mouthed monster (hatched from an egg) who lives with a wise old rabbit underground and really, really likes TV. Domo-kun is the mascot of NHK's BS2 (a satellite channel in Japan; he also appears on BS1).
"Domo" in this case appears to mean more of something (usually to stress "thank you," "excuse me," etc), implying increased politeness. "kun" is used as "Mr." or "Ms." for "younger people or colleagues". He acquired his name in the second episode, when Usajii turns on the TV for him and an announcer says "Domo konnichiwa" (i.e. "very much hello").