Your name, as shown to other users, is determined by your Profile. To edit this, go to the drop down menu next to your name (upper right) > Profile..
Your first and last names as they appear to others are shown here. If you make changes, be sure to click the Update button to save the changes.
Note: Changes to your name are not reflected in existing activity. In other words, existing activity, such as existing tasks, will still list your name as it was configured when the activity occurred, such as when the task was assigned. New activity will reflect your newly configured name.