On the current Facebook interface as of July 2015, there is a downwards arrow in the top right hand corner.
Click the arrow and then click "Settings".
You will see the name field under the "General" tab. Hover your mouse over this field and then a link saying "Edit" will appear.
Click it. Change your name and save.
Facebook checks name changes to ensure they are not offensive or nonsensical.
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Answer by Abond3
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At the top right hand corner of the screen there should say '(your name)', 'Settings', 'Logout', and then a search bar. Click on Settings. A box might pop-up while its loading saying that it may contain harmful things and if you want to still go to the settings thing. Just say yes. (I'm pretty sure its not harmful. Nothing bad happened when I went to it) it'll then bring you to the settings, where you can change your name, contact e-mail, password, your network, and a whole bunch of other stuff!
Here is a visual guide for Changing your name in facebook http://www.kapcom.com.au/How-to-change-your-name-on-facebook.html