I guess you are referring to this thing here:
You are right in that you pick the APT deb package. What you can do with is download it, and open it in Software Center - other methods are available for installing deb packages via the command line, see here.
Ubuntu (and most Linux systems), usually does not require you to download random software off the internet. Verified packages are available from the Ubuntu repositories, which can be downloaded and installed from Software Center,
aptitude etc (other packages are also available via PPAs). They also receive updates with the rest of the system, so overall this is the best method to use. For Flash just install the
By the way the NPAPI version of flash for Linux (and other non-Windowz/Mac systems) is only available as the old 11 release (excuse from Adobe is they are apparently getting rid of flash player). It works and is supposed to have security backports - if you want the new version you can use Google Chrome (or Chromium and
pepperflashplugin-nonfree package) to get PPAPI flash which still receives updates. This can be used with Firefox as well via the use of a plugin adapter - see here