To delete the dialogue with a person, open the list of all your messages and click on the X in the top right corner next to the dialog that you want to delete. Please note that the dialogs are removed irrevocably and can't be restored.
To delete the dialogue with a person, open the list of all dialogs in the mobile version, select the desired dialogue, hold and swipe it to the left. You will have three options: star, flag and basket. Choose basket to have your conversation irrevocably deleted.