Green wood is freshly cut. This means that the moisture content can be as high as 30%. Furniture and the like is generally made from either kiln dried (6% moisture) or air dried (12%) wood.
Wood will typically shrink 10% or so as it's moisture content changes from 30% to 6%. This will cause checking, splitting and warp as the moisture loss changes.
While you can turn "green" wood, as it dries your piece will likely crack.
Ideally you should used dried wood for your work. Drying wood takes $$$ (fuel to run the kilns) or time (1 year per inch for air dried). Additionally some types of wood are almost impossible to dry without splitting/checking. Slices (rounds) will almost always crack as they dry. This is because the piece dries from the outside in. The outside then tries to become smaller and *bang* you've got a split.
This is also why ends of boards are painted. Wood looses moisture from the end grain faster than from the face. If you don't paint the ends they'll dry and shrink faster than the middle of the board and you'll get a check (longitudinal crack) down the board.
If you're interested in learning about wood and how it behaves, then you should visit my web site and get yourself a copy of Wood Handbook, Wood as an Engineering Material This is an invaluable reference and tutorial. It contains charts, diagrams, photographs and discussions about all aspects of wood. Since it was prepared with tax dollars it's in the public domain.
Mark is right on the money except that there are times and places that use the shrinkage as an advantage. Some types of chairs are made using green wood. Parts are turned and assembled and the shrinkage causes the joint to lock tighter than any qlue will allow.
The other thing that can be done with green wood it to split it along the grain to all you to then shape with hand tools to end up with boards that are much stronger than any cut with a saw. The splits are then dried before it is used.
Here's a trick I learned. Turn the green wood, Like a vase. Keep the side walls thin. Pack with newspaper and let a "wick" stick out. Put it all into a paper bag, Then another. Let set for a while, like a year. and walla, no cracks. If your lucky and have the right setup. My uncle taught me this and he has made many green turnings this way.
I'm not a turner myself but have read a bit about it. Have also seen a product named PEG that you can soak green wood in prior to turning. Have seen it advertised but never really got into the specifics. You might see if you can find additional info regarding this PEG mixture.
Just yesterday, we used clamps after glue simply to take the bow out of the wood we were using, and it straightened everything out for us. Usually though, any time you lay up a project the clamps are...