There are a number of Discord features that allow you to express your identity and the identity of any character you play.
Your username and avatar
Your username is a text name by which you are known across all of Discord, and your avatar is the image associated with it.
You can change them by going into the user settings (click the cog icon at the bottom of the list of channels).
Then pick My Account from the menu on the left and click the Edit button in the main window.
This is a name by which you are known on a particular Discord server. You can change this, for instance, on a UK Freeforms server without affecting the identity you present on other Discord servers (even if you are logged in to multiple at the same time).
To change it, click the arrow next to the server name above the list of channels to open the menu.
Then select Change Nickname from the dropdown menu.