Cut the old fabrics away from the sofa, and use the shape and measurements of the old fabric as a pattern and size guide for the fabric you will replace it with. These measurements will include the material tucked under filling or framing so that you don't end up with too little fabric. Take these measurements to an upholstery material shop to help you purchase the correct amount of upholstery.
Tighten the springs beneath, and replace padding or filling if necessary while the fabric is off. If your filling is more voluminous than the previous filling, you will need to add the extra length, width and height to your measurements for upholstery material.
Cut out the new fabric according to your measurements, but always give yourself an extra inch or so of fabric, just in case. Better to have too much and have to tuck extra fabric than to have too little and have a useless section of fabric.
Iron the creases out of the newly cut fabric. This will make the fabric loose as well, and easier to apply.
Staple the fabric pieces to the sofa starting with a corner section, ensuring that the edges line up properly and that the material lies flat and unbuckled or wrinkled. However, be sure not to stretch the fabric overly tight, or any thin implement will easily slice open the sofa fabric.
Cut off any excess fabric as you go unless you can neatly tuck it down the side of the sofa discreetly so that it cannot be seen. Alternately, staple the overlap under framing for additional fabric security and longevity.
Use period-specific or decorative tacking instead of staples for more elaborate reupholstering jobs.