DO check the box "Move user and talk pages (and their subpages) to new name" if it should be moved e.g. if you rename a user. If you rename a vandal/impostor the old userpage is most likely vandalism, so delete it before (or afterwards) but don't forget to delete the redirect if you decide to move anything that should not redirect to the new username.
DO NOT usurp any accounts that have made contributions.
This is the renaming interface. The "Current username:" is the account that needs to be renamed. The "New username:" is the account that the user has requested. Write a reason for renaming in the third dialog box. If necessary, check the box that says "Move user and talk pages (and their subpages) to new name". You can check the Show block log for user before renaming. Clicking the Submit button will initiate the rename.
This is a warning prior to the rename but it does not always show up.
Check the box that says "Yes, rename the user" to attempt a rename despite the warning.
This is an error message that is shown when the account name is in use. You will need to "usurp" the account as shown below.
This is the rename interface. Notice that the "Current username:" is the target from the section above. In the "New username:" box the word "(usurped)" has been added to "Old Account" to indicate that it is being moved out of the way to make room for "New Account".
This is a warning that sometimes appears before the rename. Check the box that says "Yes, rename the user" to continue with the usurpation.
This is the notice of a successful usurpation. You have made it possible to rename "New Account" to "Old Account", which will be the next step.
This is the warning prior to the rename, but it does not show up all the time. Check the box that says "Yes, rename the user" to continue with the usurpation.
This is the notice of the successful rename. The task is complete.