Import story
To import a story published elsewhere onto Medium, go directly to https://medium.com/p/import or else click on your user icon in the upper right corner of the interface, and click on "Stories":

From the page that appears, click on "Import Story":

The import story page looks like this:

Simply paste in the URL of the story you'd like to import in the field provided:

And click on the button marked "Import":

Important Note:
As explained in our Terms of Service, you may not publish content on Medium of which you are not the owner/creator/rights holder, or for which you do not have authorization by the owner to re-publish here. If you own the content, though, and it has been published elsewhere, you're welcome to republish it here.
Editing Your Imported Story
Once you click on the "Import" button, you will see a screen like this, with the automatically-generated title of the piece to be imported pulled from the URL you have pasted into the address field:

This page gives you some instructions for updating and re-formatting your imported piece within the Medium editor. To continue, scroll down and click on "See your story":

You will then be able to edit your story using the standard Medium editor.
At the bottom of your story, the import utility will automatically add a note and link to the location the piece was originally published:

Please note:
Depending on the site from which you are importing a story, the import utility may or may not import all of the content you are looking for. It's important to verify that the imported content is what you intended and is formatted in the way that works best for you on Medium. That may be a bit different than what worked on your blog, so use your best judgement.
If the import utility doesn't automatically grab the correct part of a page, you can also just try going to the URL you'd like to import, highlight the part you want, and then copy and paste it into the Medium editor.