First of all let me say that even though the Singer manual shows how to take the tension knob completely apart--don't do it. Not unless you have to. Don't make things harder on yourself. If it's a bit dirty or you've dribbled oil in it take a folded piece of clean, lintless fabric and pass it between the tension discs. Try to avoid catching the wire take-up spring when doing this.
Also know that you must thread the sewing machine with the presser foot up. This is when the tension discs are apart. When you lower the presser foot the discs press together. If you thread with the foot down the thread won't get between the discs properly. And, this next bit is something I did wrong for a long time. To change the upper tension, the presser foot must be down. This applies to the Singer machines I use; I don't know if it applies to all vintage sewing machines.
The tension is set right when the upper and lower threads are drawn and locked into the fabric equally. If the tension on the needle thread is too tight the needle thread will lie straight along the upper surface of the fabric. If the tension is too loose, the bobbin thread will lie straight along the under side of the material. Turning the knob so the number is higher increases the tension. A lower number decreases the tension.
Bobbin tension can be changed as well. Most of the time though the upper tension dial is all that needs adjusting. I recommend trying to adjust it first. The bobbin tension is not always easy to get right.
If you happen to have a particularly old machine the tension dial might look like this next photo where it is just a knob with a large spring showing. Twist the small nut at the front to the right to increase tension and to the left to decrease.
Why is thread tension so important? If the stitching is not locked together evenly and properly the sewn item will not be durable.
You can do a scrap test of your tension. Thread your machine with a size 60-70 thread (basic thread), make sure you have the right size needle in, and sew diagonally across a double thickness of muslin. Holding the stitching at both ends pull evenly until the thread breaks. If the tension is right the upper and lower threads will break in approximately the same place. If only the bobbin thread breaks, the needle tension is too loose. If only the upper thread breaks, the tension is too tight.
I had to do this test a few times to get my tension set right. Once you do get it set correctly make a note of it. If you have to change the tension for other sewing you can easily put it back for general sewing. And, of course, just look at the stitching and see how it appears to you--if you've sewn much you have an idea of what's right and what isn't.
Tension is not the only thing affecting a stitch. Make sure the needle and thread are the correct size to be used with your fabric. It's important that the stitch length is right too--which I'll cover next week.
Oops! I mentioned bobbin tension but forgot to tell how to change it. The following photo has arrows pointing to the screw on the bobbin case that changes its tension. There is a spring under where the thread passes. Turn the screw to the right to increase tension and the left to decrease. A tiny amount changes the tension considerably so go slowly with a little at a time.
A general rule of thumb is: loops on the upper side can be corrected with the under tension; loops on the under side can be corrected by the upper tension. This is general though and a lot of times the upper tension can correct most problems.