There's not a direct "upgrade" path from Ubuntu to elementary OS. You'll need to install elementary OS, then copy your files over and re-install your applications.
There are a few ways to do this. You could:
- Move your content to another partition, and then replace Ubuntu with elementary OS. (See above answer for details.)
- Back up your files to a third-party service such as Dropbox, Google Drive, or SpiderOak. Then, completely overwrite Ubuntu with a fresh install of elementary OS, and re-download your files.
- Install elementary OS alongside Ubuntu. Copy your files from the Ubuntu partition to the elementary partition. Once you're ready, delete the partition containing Ubuntu.
The installer should be able to handle a clean install or a dual boot setup for you. You'll see a message like the screenshot below. (It will say "Ubuntu" instead of "Windows 10", though! 😀)
See also: the Dropbox for elementary OS installation script.
Image source: LinOxide