If you have never edited your YouTube channel URL and your URL looks something like this (http://www.youtube.com/channel/UCY3GS9HK953AaPiTRnTMTXw) then you can change that URL by going into your Advanced Settings area and clicking on the Create Custom URL as shows in the screenshot below:

To get a custom URL for your channel, your account needs to be in good standing and meet a few other requirements:

  • 500 or more subscribers
  • Channels is at least 30 days old
  • Channel has uploaded a photo for the channel icon
  • Channel has uploaded channel art

OR You can also qualify for a custom URL by linking and verifying your official webpage with your channel And G+Page.
If you receive an error message saying "You are not eligible to perform this action" it means you cannot edit your URL if you have already done it once or if you have integrated your channel with Google+. Unfortunately there is no way to undo this because YouTube stopped offering custom YouTube channel URLs since they started integrating with Google+ which allows you to change your display name to match your Google+ name.
You can change your channel display name by changing your G+ display name. All you have to do is go to your G+ page and click on your name to change it. You will see the change reflected on your channel right away!