Careful with the nomenclature. An attachment is anything that goes in a slot. It could be an image, bounding box, etc. There is a special type of attachment called "skin attachment". This is the one that has a puzzle piece icon. This are one of the trickier parts of Spine, but I'll give explaining it a go.
The skin attachment provides a level of abstraction for what is actually attached there. It is a named placeholder. What is actually attached comes from a skin. Switching between skins changes what images are used for the skin attachments. Skins allow different looking characters to reuse the same skeleton and ALL the animations.
To use skin attachments, create them under a slot. Eg, under the eyes slot, create eyes-open and eyes-closed skin attachments. Next create a new skin. When a skin is visible, you can put images in the skin attachments. Put in images for eyes-open and eyes-closed. Create another skin and notice that the skin attachments are empty. Put in images for eyes-open and eyes-closed for that skin. Now when you switch skins, the images in the skin attachments change appropriately.
This way of doing skins may seem a little clunky at first, but it is much more powerful than a more naive skin solution which might just directly change the images that are in slots. Imagine you have one skeleton and two skins, a green goblin and a purple goblin. In an animation, you want to have the character blink. For the "eyes" slot, you simply show the correct skin attachment, either "eyes-open" or "eyes-closed". This way the green goblin uses his eyes open/closed images and the purple goblin uses her eyes open/closed images.
If the skin system didn't use skin attachments and instead it just changed the images that are visible, what would you do to blink in an animation? You would have to show either the green OR purple eyes closed, and then it would look wrong for the other skin. To fix this you would have to have an animation per skin, which means you aren't reusing the animations at all!
Note that skins are not meant to be used to change what weapon is in the characters hand or what hat she has on. For these, just use slots with regular images. You will have to manage setting which images are visible on which slots in your game code. Skins are for when you have multiple characters that use the same skeleton and you want to reuse all the animations.