When a browser loads up a note, it will also receive an ETag from the server. This ETag will be a hash of some unique information about the note in Evernote's DB. An example of an ETag would be to hash the note's DateUpdated value and the note's ID. Whenever the browser sends an update request to the server, it sends the most-recently-received ETag with it. The server then recalculates the hash on its own and compares the server-sided ETag with the user-provided ETag. This definitively proves that the browser either did or did not have the most up-to-date information at the time of sending an update request.
For example, if you changed the note on your phone, then hopped back on your computer (which has the tab open to that note from earlier) and tried to edit that note, the server will quickly determine that the browser's update request was out of date. At this point, the devs can do whatever they want: force a refresh of the user's browser, give the user a prompt that asks them if they want to overwrite, etc.