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Do you mean the little box to the left of each title? You can use those to mark titles and then "Delete Marked" (like when you have multiple versions of a book on your WL and you get one) or "Convert Marked", which will move the titles to your Reminder List.
Books you receive are placed on your bookshelf, but on Hold, so that if/when you're ready to repost them into the system they're already there, all you have to do is make them available by checking the box next to the title and clicking the Post button. Books already posted have a green check, books on hold have the red circle/slash over the green check.