No, the (earth) worm is not considered to be an insect. It is rather an annelid: a group of animals consisting of some types of worms, leeches, and a few other (but not all) bilaterally symmetrical organisms. read more
As for insects, many larvae have the body shape of a worm. Since the term is solely based on body features, I think that any animal that looks like a worm, IS a worm, in the language of the general public. read more