Step 1: Making the covers
To create a pattern for another chair, the easiest way is to use cheap cotton muslin fabric. Drape it over the chair's seat, and mark it with sharpie where you need to cut. Allow it to drape down the sides and then give it an extra two inches underneath the bottom of the seat to cover it completely, then another extra inch to allow for a seam for the elastic (total three inches all the way around, beyond the bottom of the side of the seat). Sew a sleeve around the whole edge except for a 2" gap, then thread a strong elastic cord through the sleeve. Pull it tight, the seat cover fits over the seat like a shower cap, with elastic around the lip to hold it on snugly. Tie off the elastic for a snug fit.
To create a pattern for the back, drape the fabric over the back and mark it at top center, then all along the sides. When you cut, leave an extra half inch all the way around for the seam allowance.
The back and front covers of the seat back are sewn together along the top and sides, leaving the bottom open to slip it over the back. Then you can close it at the bottom either by hand sewing it together, or using stick-on velcro (or zippers if you're motivated and you're good at sewing) along the edges if you want the cover to be easily removable.