How can I play .mp4 files?

Welcome to LQ, OldUnixUser.

Most distros have a default video player. If you let us know which distro (and version) you are running, and which desktop environment you are using, then that will provide us with better information to answer your question.

You need the media codecs installed. Because they are patented in the USA, distros connected with companies that are suable there (e.g. Fedora, OpenSUSE) leave you to get them yourself. If your media player says it can't play mp4, you may be in that situation. Searching on-line for something like "fedora media codecs" will point you in the right direction. Otherwise, tell us your situation and we'll do our best to help.

As hydrurga pointed out, it does help if you display your distro as we do.

or you could install VLC, how you do that is different from distro to distro, but VLC plays nearly everything without the need for additional codecs.

I am using SUSE LEAP 42.1 with all current updates. I am using the current version
of Firefox.

When I attempt to play an .mp4 file (that I have received via email), the Dragon
player opens but will not recognize or play the file.

What is a VLC?

My background: BS in Software Engineering. Retired for 10 years.

I programed SCO and SUN Unix in various languages for over 30 years and taught
College courses in both of those operating systems as well as all of the Unix
shell script languages...also ADA, C, Perl, and Java. But I am new to Linux and
using Unix/Linux and very new to using it as my primary internet OS....went to
it after screwing everything up by loading that WIN-10 piece of garbage.

$ avprobe file.mp4

or

$ ffprobe file.mp4

Chances are that it's h.264 codec wise, so you'll need to install the x264 stuff.

$ sudo apt-get install x264 libx264

You probably only need ONE of those and of course SuSE is YUM based, so different package manager and probably different package names.

https://software.opensuse.org/package/libx264

As I said, OpenSUSE is one without the codecs and the repository version of VLC is no different. You can get VLC with support built in, though:
https://en.opensuse.org/VLC_media_player

Or you can get the codecs separately and use any player, which probably makes more sense:
http://opensuse-guide.org/codecs.php