Shipping profiles can only be deleted if they have no products associated with it. So, if you see that there's products associated with the shipping profile, you first need to go through your listings and swap the shipping profile that it is associated with. You can see what shipping profile it belongs to on the Item Management pages.
Changing the Shipping Profile on a product
Simply click "edit" under the product title, then scroll down to the Shipping Profile section. You can either select another of your existing shipping profiles, or you can create a new one.
Once there are zero items associated with a profile, you'll be able to delete it via the Shipping Profiles page in your store's admin panel.