As my man Webster says, smithereens are fragments - usually little bits of something that has just been smashed or blown up. So what's a smithereen?
It comes to us from Gaeilge (Irish Gaelic), in this case the word smidirín. -ín is the Gaelic diminutive suffix, and gets anglicized as -een, like in Colleen, from cailín ("young girl"). Smidirín is the diminutive of smiodar, meaning "fragment".
What Webby calls a "smither" is probably just an anglicization of smiodar.
Example use: "We dropped a lit quarter stick of dynamite in a can of dog food and ran like hell. It blew the thing to smithereens. I found pieces of the can a quarter mile away."