It sounds like a weighting problem. Select some of the points that you don’t want to be moving, and open up the component editor.
(Window>General Editors>Component Editor) Select the tab that says “Smooth Skin” and you’ll get a list of all the joints that are influencing that vertex. Zero out the ones you don’t want. For example if you move the arm and some points move on the hips, then for those points you want to find the influences that it’s getting from the arm, or fingers or whatever and set them to zero. If there are a lot of verts being influenced, or the verts are being influenced a little bit by lots of joints, you could try Skin>Edit Smooth Skin>Prune Small Weights. Set the value to like .005 or something, and if you still get influence then try to raise it a little bit.