Batting or "stuffing" for the less sophisticated among us is the jam in the doughnut. The doughnut is going to be yummy no matter what you put in the middle of it, but some jams taste better than others. And so it is with batting the middle of your quilt is often an after thought, but it could add some flavor.
How heavy do you want your finished quilt to be? The default batting for most quilters is the good ole' Warm n' Natural. It is a good all around cotton batting, but not always the best choice. Standard weight cotton batting can get heavy in a large queen or king size quilt, now imagine that giant quilt after it gets wet (yes you will have to wash it at some point) cotton weighs substantially more when wet so moving the quilt from the washer to the dryer could give you a hernia.
Synthetics (a.k.a polyester) have the advantage of having a lighter weight and not putting on extra weight when wet. But in smaller projects that lighter weight often results in poor "draping". A table runner or wall hanging won't display as well with lighter batting.
Do you want a puffy quilt? If so you want a high loft batting. Wool and high loft polyester give a third dimension to any project. But remember loft and weight are two different things.
How close are you going to tie/stitch your quilt? Stability is a reference to how far apart you can stitch/tie the batting and it won't shred or separate on you. This is important to keep in mind because if your batting falls apart inside your quilt it may still be pretty to look at, but it will suck to sleep under.
Not all batting is created equal. Some brands are better than others. Synthetics and blends especially can vary greatly between brands. Often as with most things in the quilting world you get what you pay for. So don't be afraid to ask for samples if you want to try something new or unsure of what you are getting.
There you have it, things to consider when putting the jam in your doughnut.
"You took the #$%*& jam outta my doughnut, Tommy. You did." - Turkish (a funny name for an Englishman)