Please note that the UI of VSTS has recently been changed. I'm updating my answer to reflect the latest changes but keeping an older copy too!
Update: As of 17th Oct 2018, on Team Services:
- Account Home - Click on the Azure DevOps icon in the upper left corner.
- Organization Settings - Click on this link in the lower left corner
- Process - Click on Process in Boards Group in the list of settings
- Select your process
- Select the Work Item Type you want add the new state to
- Click on States Menu item on the top
- Click New State button
Adding a Gif below
For those who are still on the older UI , on Team Services...
- Account Home - Click on the team services icon in the upper left
corner.
- Process - Mouse over the Gear icon and choose Process
- Choose the process you are working in
- Click on Product Backlog Item type
- Click on the States tab (between Layout and Rules)
Click the New State link
- Account Home can be accessed by clicking on the Azure DevOps icon (previously Team Service) in the top-left corner. This page is also the home page of you account and can be accessed by directly jumping using the URL. For example: youraccountname.visualstudio.com/
- Process can be accessed from the UI as mentioned in steps above or by using the direct URL youraccountname.visualstudio.com/_settings/process
Hope it helps!
Thanks Microsoft for making it available!