Publishing is a way to lock in your current setup of your workflow for production use. This will create a new published version of it, which will be immutable and will be the one you will see when you run your workflow from the dashboard.
How to publish
After you are done editing your workflow and testing it, your next logical choice is to publish it. You can achieve this by click on the Publish button on the top right of the Editor. This will take your current setup and create the published version, then it will redirect you to the dashboard so you can run your published workflow.
Next time you go back to the workflows tab on the dashboard, you can see that your workflow is published if you see the Run button on it.