If you look at your machine, the feed dogs are the two rows of "teeth" underneath the presser foot that feed the fabric. Without them, or if you drop the feed dogs so they are underneath the surface of the machine bed, the fabric does not move by itself. If you don't move the fabric, the machine will sew in the same spot over and over.
The bottom of a darning foot is usually shaped like a small doughnut -- a small circle with a small hole in the middle. Often it is made of plastic, but it can also be metal. If you examine a darning foot, you will find that there is a spring encased in it somewhere so that you can move the foot part up and down on the shaft. This is what provides the "hopping" action as you sew. With every stitch, the foot "hops" to release pressure on the fabric, allowing you to move the fabric in any direction you want. (Feed dogs, in contrast, move the fabric only forwards and backwards.)