Originally (in the slang of the 1780s) used of a person, but also used at times of, for ex, furniture, like that overstuffed, low sofa in your aunt's parlor. It can have connotations of cute or comfortable, but can also be an insult. It's no longer a common word, but it still survives in England, as opposed to the older and obsolete noun it came from, fub or fubs, being a chubby person, especially originally a child. Its origin is, not surprising for slang, unknown.
The detective was a fubsy man with a vague manner and a wrinkled trenchcoat, looking for all the world like a slightly squishy Columbo.