Despite Webster 1913's definition of "spatchcock" as just a variant spelling of "spitchcock" (which it defines as "an eel split and broiled"), there are other definitions of the word. A.Word.A.Day says that to spatchcock is to insert or interpose something in a forced or awkward manner, or to split open a fowl for grilling (most likely the source of the more general meaning). It says the word is of unknown origin, but possibly a variant of spitchcock. A site with a Victorian-era recipe for "Spatchcock" (which starts out "with a large sharp knife, split the chicken or poussins through the back, close to the backbone and beat flat with a steak beater") says "The word spatchcock comes from the dish - a happy rooster one minute and a rather flat-looking grilled chicken the next - all done with great despatch." However, A.Word.A.Day notes that a derivation from "dispatch cock" is "etymologically not confirmed" and the American Heritage Dictionary agrees that the "spitchcock" origin is more likely. The OED says the word is Irish in origin and later adopted by the Anglo-Indian community, but does not say what words it stems from. At least one site says that "spitchcock" may have applied to fowl before it was used to mean eels, and that the word might derive from "spit cock" -- that is, cooking the rooster on a spit.
Modern recipes, unlike the simpler 19th-century version above, seem to cut the backbone out completely, presumably to flatten the bird out even more. (Amusingly enough, the French term for the same technique is "en crapaudine" -- in the style of a toad. I suppose a stretched-out toad bears some resemblance to the flattened chicken with limbs sticking out in different directions.) The benefit of this opening and flattening technique seems to be that the fowl cooks faster and more evenly, particularly when it's grilled or roasted. Several cooking sites apply the word to the splitting-and-butterflying preparation of other fowl, such as quail or grouse, but it never seems to be applied to anything but birds (with one peculiar exception, a site touting "the Sausage Spatchcock," which seems to be so called because it's a large spiral of one long sausage which is as flat as the spatchcocked fowl and provides such advantages as "you do not have hundreds of sausages to manage; ideal for parties it can be sliced according to appetite." Usually, "spatchcock" as a noun refers to the chicken prepared in this way, but there are other usages: Gamefarm.com.au says a spatchcock is simply "a spring chicken processed at 3 weeks of age." Other sites say it can apply to chicken up to six weeks of age. (Sneff, Australian resident and expert on things culinary, adds that "I can confirm that in Australia at least, spatchcock refers not only to the preparation, but also the retail name of baby chickens - of weight between 300gm and 600gm.")
The majority of the sites I can find that use this word in any of its meanings are either British or Australian; the word doesn't really seem to have made it into American or Canadian English.
Sources:
http://wordsmith.org/awad/archives/1202
http://www.gamefarm.com.au/spatchco.htm
http://www.hwatson.force9.co.uk/cookbook/recipes/starters/spatchcock.htm
http://www.nakedwhiz.com/spatchdef.htm
http://www.nakedwhiz.com/spatch.htm
http://www.bartleby.com/61/94/S0609400.html
http://www.tiscali.co.uk/reference/dictionaries/difficultwords/data/d0012066.html
http://www.hyperdictionary.com/dictionary/spatchcock
http://www.thefreedictionary.com/spatchcock
http://www.harpersfood.co.uk/pages/recipes/r2001nov.htm
http://www.geezergourmet.com/tech_art11.html
http://www.grilling-recipes.com/chicken/r212.htm