I tried myself on doing a rig for a low poly crab and somehow I changed some settings without realising. Now I'm not able to move the bones except of the base. Normally, if I select one, it looks like this:
If i try to move it now this happens:
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A bone is defined by two points, known as the head and tail of the bone. When you select one of these points you only move that point, the bone/s defined by that point then stretch to fit the opposite point.
If you want to move both points of a bone together you select both points. Clicking the body of the bone will select both points for you.
While I don't think it applies to you, you can lock the points of a bone in the bone transform properties, so that they can't be moved in edit mode. If this is the case, it is a matter of turning off the lock.