When you first sign up to create a Yahoo email account, one of the questions you'll need to fill in is your name: this information is used mostly when you send messages from your Yahoo Mail account from a web browser, and will show the name you picked to any of your email recipients. The first name you enter will also be shown whenever you sign in to Yahoo, as shown on the screenshot. If you ever want to change your name in Yahoo Mail (also known in email parlance as your "display name"), it will take a few tedious clicks but is easy to do - not as seamless a changing your name in Hotmail, for example - Yahoo, if you are listening...
Caution: Because Yahoo Mail and your Yahoo profile are connected, changing your name in Yahoo Mail will affect all other services that use your Yahoo login for authentication. When you go to the Yahoo homepage, for example, the first of the name you're about to change for your email account will be displayed.
Finally, scroll all the way down and click on the "Save" button: you can close the browser window / tab to return to your emails. From this point on, your new first name (if you've updated that portion of your name) will show at the top of web pages like Yahoo News - reload your inbox and you'll see the change.
Tip: changing name in Yahoo Mail only affects messages you send from Yahoo Mail inside a web browser. If you are using a desktop email program connected to your Yahoo account, you will need to update your name in that program as well - this is typically done by customizing your email account's properties.