My mother has swelling on her left shoulder almost near the neck and it is going towards her back. She has no pain. What can this be?

Needs evaluation. This needs to be properly evaluated. Your description fits a simple lump of fat called a lipoma, or cancer spread from somewhere else in the body to a lymph node. You should encourage and support her to seek proper help. If it is nothing, it can end up being ignored. Or, getting prompt treatment might be life-saving.