To understand how attachment points work with mesh items, you must understand how mesh items work on a basic level. The two basic types are rigged and unrigged, and while there are subtypes, those are not important to know for what you're trying to do.
The idea is, that when you attach a rigged item, it will always attach to the same part(s) of the body, because it is rigged. Rigging is basically adding extra data to a mesh item before upload that tells it how to fit on the SL body, and how to move with it. Rigging cannot be disabled; if you wanted an unrigged version, you'd have to ask the creator to upload an unrigged one if they haven't already.
Unrigged items are simply that - not rigged. They did not have that extra data added to them before upload, so they don't know where on the body to snap to when you attach them, and they do not bend at all. For these items, they need to be told where on the body to anchor to, via attachment points. All non-mesh objects work like this as well.
Long story short, no matter where you attach a rigged item, it will always go where the rigging tells it. If it changes position whenever you attach it to a different spot, then it's not rigged. I may have been a bit redundant, but it's all in the name of clarity.