How can I write a story?
To write a story on Medium, you'll need to set up a user account and sign in to Medium.
Once you're logged in, you can click on "Write a story" in the upper right corner of the interface:
You can also click on your profile avatar/icon, and click "New story" on the drop-down menu:
One last way to start a new story is to go directly to http://medium.com/new-story.
Whichever method you choose, when you begin a new story on Medium, you will see the Medium editor, which looks like this:
In its most basic form, a story on Medium consists of a title and a body text. (And, of course, it can get quite a lot more complex than that.) You're not required to give your story a title, necessarily, but you do have to include a bit of text before you'll be allowed to publish (i.e., you cannot publish an empty story).
For now, we'll keep things simple. 
See how easy that was?
Anyway, let's not get hung up on content for right now. To publish your story on Medium, click on the "Publish" button in the upper right corner:
You'll get a drop-down box to help you get everything organized:
Note: Depending on the type and length of the content you've included in your story, Medium will try to guess some tags that might be helpful for you to include. Some of these guesses are better than others. In this case, let's not worry about it too much, but you can always delete inappropriate tags by clicking the "x" next to them and add new ones. (You can add up to five tags on desktop and three using the app.)
Let's just click on "Publish," which will make your story live on Medium.com.
Your published story will look something like this:
At the top of the screen, you'll see the name associated with your account—in this case, "New Writer." Underneath, it says, "Just now," to indicate that the story was just published. Over time, this will change to reflect the publication date. Below that are the title and text of your story. Further down, you'll see any tags associated with the story, followed by the "heart" button for people to recommend your story, and the "word balloon" icon which allows people to leave responses.
If you are reading through your story and spot some errors, you can always click on the "Edit" button at the top to modify it, and then re-publish it with the changes, using the same steps described above.
Story URLs
By default, new stories are published at unique URLs that are structured something like this:
https://medium.com/@newwriter/this-is-my-story-148ecc9a7bca#.kw43tyyqm
First comes "medium.com" followed by your @username and then the first few words pulled out from the title (if you have one), or from the body of the story, if there is no title. Following that is a string of characters, appended to ensure that each story has a unique address.
Later, when you add stories to publications, the @username section of the URL will be modified to reflect the name & URL of the publication it's associated with.
Once your story is published, the URL is final. The story URL won't change if you edit the story title, so it pays to plan your title well. Alternatively, you could copy and paste the story into a new draft and set up a better title. Publishing it will create a new story with a new URL, and you can delete the old one.