From Kindle app, you can only archive books, not delete. To archive from app, on right side of screen, swipe left to right (or right to left) and a popup will let you archive. You can delete samples that way from iPad, though.
If you want to delete a book, you have to go to Amazon's website, log in, go to Manage Your Kindle page, drop down the Actions menu and choose Delete from library. If you do that, it will be as if you never bought the book. So if you change your mind, you will have to pay for the book unless it's free.
