For the federal return, you only have to send in the 1040X, unless the change involved needing to add another schedule or form. In that case, you would include the added form.
Not every state has the same set of rules. However, if the federal amend doesn't affect your state refund or liability, you would not send anything in. If it does, you would send in the state forms (dependent on the state) along with the 1040X.
Here is a little background on filing an amended tax return. IRS Guidance on an Amended Return
