Let me preface this post by saying that not all chairs are created equal. This awesome vintage chair, that I either found for free or inherited from someone, had a dingy upholstered seat but was super easy to take apart and figure out how to put back together. The cushion effect was gone and when I opened it up, the padding was crumbling away and the fabric was disintegrating.
For years, I had this subtle chenille fabric with a blue/tan pattern that I got in a clearance bin at JoAnn Fabrics for less than $5. I bought foam pad at Michael's with my 40% off coupon ($6.50 for 2-pack) but only needed one of them. And since I didn't have one in my arsenal of tools, I also picked up a stapler gun via Amazon.com ($12.50).
Removing the existing seat was easy. You just unscrew the screws that are there and it pops right off. Throw out the padding and fabric and clean off the wood base. Cut your new fabric with a good 6 inches on each side so there's room to pull it over the foam. Cut your foam to fit the wood base and give it 1/4 inch more on all sides so you don't feel the wood's edge once it's done.
Now you're ready for the stapler gun. Even the light-duty versions of stapler guns are fierce, so be careful. On a good size workspace, lay out the foam on the wood base with your 6 inches on each side. Some folks glue the foam to the wood but that's up to you. I didn't and it worked just fine. Pull tight on one side making sure you're not pulling away from the even distribution, and staple once in the middle. Then go to the opposing side and staple once again. Do the same to the other sides until you have the center points firmly taut. Then work your way from those center points and staple as far as you can to the corners, always keeping the fabric tight but making sure not to pull fabric away from the opposing side's allowance. You will have more fabric than you need at those corners once you reach them, so cut some away carefully so you'll be able to neatly tuck the excess and make a nice, smooth corner staple for each (see how I did it by bunching it up a bit).
Once you're done stapling, it's ready to go back on the chair. Use some blue painting tape to secure it to the chair so when you flip it upside down, it won't fall away. Flip your chair and screw those same screws back into the wood base. Viola!