If you want to edit your current username or email address, all you have to do is go to your settings and edit the name that is there. It has to follow the same rules as creating a new username (for example, it can't already be taken). Remember to save your changes on the green button on the top right of the area.
If it is not changing, it means it is already taken by another Duolingo account. Try changing the name again, by adding unique letters, punctuation or numbers to try to make it unique and save again. Refresh the page to see if the change was successful.
Warning! When you change your username/email, your previous ones will be available for anyone to take, so please consider this when making changes or when announcing a change publicly. Staff will not be able to recover an old username for you if someone else "steals" it.