Edit a published story
To edit a story you already wrote on Medium, navigate to that story's page while signed in and click on the "Edit" button in the upper right corner.

Your story will go into editing mode, where you can add or remove text, formatting, images or other embeds.
When you go into editing mode, the "Edit" button turns to a "Publish" button, which you can click again when you're finished editing.

The browser tab title will also change to include the word "Editing" in front of your story name:

You can now edit as you would normally when writing a new story.
When you're done, click "Publish," which will open a drop-down box where you can confirm your update by clicking "Publish":

You'll see a note at the top of the screen telling you that your changes have been published, and will be returned to the published story page.