Measure the surface of the chair seat (and chair back, if it requires re-covering as well), side to side and back to front.
Add 8 inches to each measurement. For example, 18 by 20 becomes 26 by 28.
Count your chairs.
Determine exactly how wide your fabric is. Most fabric is either 36 or 45 inches wide, but most upholstery fabrics are 54 or 60 inches wide.
Figure out how many pieces you can put side by side (on 54-inch fabric, you can put two 26-inch pieces), making sure to take the fabric's design into account. A vertical design will look funny going side to side and vice versa; if this is your first time, you're best off with one that goes both ways.
Figure out the closest yardage measurement to your vertical (up and down) measurement. Fabric is sold in yards: A yard is 36 inches, so 1/2 yard is 18 inches. A 28-inch piece is just over 3/4 of a yard (27 inches), so will require 7/8 of a yard, or 31.5 inches.
Multiply this by how many chairs you have and how many pieces you need. For example, if you have six chairs and are re-covering the seats only and two pieces will fit side by side on your fabric, take the vertical requirement and multiply by 3 to get your total yardage requirement.
Choose your fabric (remember, these chairs get a lot of wear and tear, so pick something reasonably durable).
Take the chairs apart. This usually requires unscrewing the seats from the frames (if they're stuck, apply a little WD-40 with a cotton swab).
Remove the old fabric or vinyl - you may need pliers if the staples or tacks are old.
Double-check your measurements to make sure the fabric pieces are large enough to wrap around the seat bottoms on all sides, and then cut out your new chair covers. (If they're not too disgusting, use the old ones as a pattern.)
Place your new cover on your work surface, wrong side up.
Place your chair seat on the new cover, top side down.
Hold the two together and flip to check that the fabric is pointed in the right direction; adjust if necessary; place the seat and cover back back down on the work surface, as before.
Pull the fabric up on one side (keeping the design straight, if applicable) and staple, working from the middle out to the corners.
Pull the fabric up tightly on the opposite side and staple.
Repeat with the remaining two sides.
Repeat with each chair.
Reattach the seats to the chairs.