Red - The main framework of the cow - built out of 2x3's
Yellow - Support boards stapled to the inside of the cardboard
Green - Basic outline of cardboard
Blue - Location of tail and udders

I'll take this step by step.

1. Body - The cardboard body is a piece of cardboard about 40-44 inches wide and about 6 or 7 feet long. It is rolled into a barrel shape (a small old tire or something works good to put inside when rolling the cardboard), then duct taped like crazy to stay in its shape. Use the tire or whatever to then trace 2 circles out of cardboard to fit in each end of the body. Then take a couple of thin wood strips and staple them to the back of these circles across the grain of the cardboard to stiffen them. Put the front circle inside the body, about 1/2"-1" in, then tape it real good all the way around so it stays in. Leave the rear circle out for now tho! Here's a crappy picture on putting the end in.

2. Frame - There are a set of 2x3 rails that run front to back inside of the cow. The legs (about 20 inches long) nail into these rails from inside of the cow ( have fun nailing from inside the cow). The neck comes down thru a hole in the top of the body and is toenailed into the side of one of the rails.

3. Head & tail - The head is built out of a showbox and another smaller box duct taped together. Nail this to the top of the neck from the inside of the box (before you tape it all together) with some wide-head nails (like roofing tacks). Take the second circle that you cut out for the rear of the body, and cut about a 1" diameter hole near the top of it for the tail to come out of. Get a piece of hose or something that will work as a tail, put it thru the hole about 4" and duct tape it a LOT to get it to stay and not rip out. Then fit the rear circle into the body same as the first and duct tape it in place all around.

3. Skin & butcher it - get some white butcher paper or other heavy paper and wrap the cow using scotch tape to secure it. Wrap the head, neck and legs, and crumple up some little piece of paper into triangle-type ear things and tape them on. Then blow upa rubber glove to a good udder-sized balloon (making sure the fingers are sticking out nicely), tie it shut, and duct tape it to the bottom in the appropriate place.

4. Detailing - Use white paint to cover up any patches of duct tape showing, then use some flat black paint and paint on the spots. Be sure to get two good spots on the nose for nostrils. Cut out two holes in the "Brain" for eyes and get some of those glowsticks (the kind you bend and they glow). When the cow is deployed, just crack the lightstick and shove it in his brain thru the eye holes.