Publishing an App comes after setting its Security and means that you are ready for the App to be visible in the Launchpad and used by Users. It represents the final step in the App-build process.
To Publish an App, you must first be in the 'Publish' section. Once there you will have the option to publish your app.
The two indicators, 'Is Description Set?' and 'Is Title Field Set?' are there to help ensure the basic coherence or your App. If they are not both green, you should consider amending this prior to publishing. However providing you are happy for your App to be published and go 'live', simply toggle 'Is Hidden?' to off and hit 'Save' to commit the change. Your App will now be visible from the launchpad to those Users that have been granted access in the 'Security' section above. (See for information on your App's Security)
Once you publish changes to an existing app, users will receive those changes within 2 minutes of being on the platform. Some navigation will be required in order for the user to see changes that affect the current page. (e.g. if a users is viewing a list report which has been changed by an appbuilder, then that report will not change without the user either refreshing the page or navigating away and back (or changing or opening a record and coming back)