this is one of those problems that could be anything. I am going to give you a few things to try...
First thing, clean, oil, and grease your machine. I mean, if you are going to try to fix it... it is easier to start with a clean machine. Get a piece of soft light weight cotton fabric, soak it in alcohol and run it around the tension disks. This is part of the cleaning process that we tend to neglect. Make sure to get in between the disks.
Once you have given it a once over....
change bobbins, sometimes there is a little nick on the bobbin or it didn't wind just right which could be a function of the spool of thread. And sometimes the thread on a spool doesn't get wound with a smooth tension from the manufacture so give another spool of thread a try.
changing the needle or my favorite thing to do is use that "strawberry hanging from the tomato pincushion". The strawberry is filled with emery which is abrasive and will smooth out any nicks also referred to as 'burrs' and sharpen the needle at the same time! check to be sure the needle is straight.
Make sure you have the bobbin in the correct way and the thread in the tensionhttp://www.sewusa.com/Threading_Diagrams/Threading_Pages/Singer_Sewing_Machine_Threading/singer_201_threading_diagram.htm
If none of these things work, try adjusting your tension knob, a fraction of a number at a time. Sometimes these old machines springs inside the tensioner just loose its spring and need a little gentle turning of the dial.
if not.. take it to a repair shop. Usually for the price of a cleaning they will do all the adjusting needed. I would never give up a 201! unless someone gave me another 401! LOL!