Hi, I'm just testing the Linux waters. I have two main questions. First what are some key advantages and disadvantages of the few most popular Linux desktop flavors. E.G. Ubuntu and Fedora and the next couple most popular ones, whatever they are? Second do apps made for one run on them all? Thanks for your help!

Just try as much as you can. You will notice the difference pretty soon. Some distros are easier than others and some are used mainly for desktops and servers and some are specialty distros to function as a router/firewall or a rescue CD.

Most distros will have the same apps, others will have more than others and some distros like fedora will require extra repositories for non-free software.

Visit http://distrowatch.com/ for many flavors and info on these distros

Package management and default desktop are the primary differentiators. Ubuntu uses .deb with Unity, Fedora uses .rpm with Gnome, Slackware uses .tgz with KDE, etc etc etc.

Of course, there are respins of most of these distros to change the desktop (Kubuntu is Ubuntu w/ KDE, Lubuntu is Ubuntu with LXDE, etc).

There are, of course, myriad other differences, but these are the biggest that usually influence someone to choose distro A over distro B.

Most of the popular distributions have what are called software repositories -- these are servers containing software for that distribution. Software is installed from these repositories by either using something like a "Software Center" or by simple commands like "yum install firefox" or "apt-get install firefox". Most of the major distributions will have virtually all the software you need in these repositories -- in most cases they'll have everything you need at least when you start out.
Outside of the repositories there is software like Google Earth as an example which is packaged for, say, Debian based distributions (this includes Linux Mint and Ubuntu) in a file ending .deb and for other distributions in a format called RPM or simply as a tar.gz -- for installing applications like this it is usually a simple matter of using a search engine or the help pages for your distribution to find out which file you would download and how you would install it.
The first place you should look for software though, in most distributions, is in the repositories for that distribution using the package manager or "Software Centre".

To take the second question first, any Linux application will run in any distribution. But there are several different packaging systems (like msi files in Windows) in use: rpm, deb, etc. Also, Linux software is integrated. If 2 windows programs use the same dll file, they will each have their own copy, but no matter how many Linux applications you have using a library, they'll all share the same copy. That means that a package has to specify its dependencies and the installer has to check they're installed. That can lead to problems even if 2 distros use the same packaging: Fedora and PCLinuxOS both use rpm, but library A may be in package X in Fedora and package Y in PCLinuxOS. That's why we say "get your software from your own distro unless you know enough to sort out any problems." Luckily, most major distros have a huge range of software:
http://pkgs.org/
http://linuxappfinder.com/alternatives

Now characteristics.

1. Unlike Windows, you can choose your GUI. Most distros offer several, but it's a good idea to use the one they offer as a default, since that gets the most users and is least likely to have bugs. You can choose
Unity: makes your computer look like a smart phone: Ubuntu only
KDE: lots of eye-candy, very configurable: PCLinuxOS, Mepis, Korora
Gnome: quite fancy but less demanding on the hardware than KDE, a bit more tablet-looking: Korora, Parsix
Mate: more traditional in appearance and performance: Mint
Xfce: ditto: Manjaro, SalineOS, ZevenOS

2. Distros vary according to how long they test new programs and so how breakable they are, from bleeding-edge to enterprise quality.
Bleeding edge: Manjaro, Korora (much less so)
Ultra-stable: Mepis, SalineOS

3. The frequency of release also varies. Rolling-release distros change constantly, so you install once and update once a month.
Rolling-release: Manjaro, Mint (Debian edition) PCLinuxOS
6-monthly: Korora, Mint
annual: Mepis, Parsix, SalineOS, ZevenOS
alternate years: Mint {long-term support version)

The ones I've mentioned are just those I think are good-quality, easy to install, and well-off for software.

1) Popularity itself: If you know how to use google, and try that whenever you are confused over an aspect of use, the more popular your chosen distribution, the more likely it is that you will find a distribution specific answer exactly fitting your question. So far as I understand, Ubuntu is still the overwhelming winner in that comparison.

2) Cutting (bleeding) edge: If it matters to you, distributions vary a lot in how rapidly they adopt new developments. Too fast and you are beta testing software, not using safe software. Too slow and you are waiting for that great new feature you read about that everyone else has.

3) Long update without reinstall: If it matters to you, and two actually different but related questions (and I'm not sure myself where you find the answers):
a) How long will you get security and other critical updates for this basic version before it is dead?
b) Can you jump all the way to the next major release by the update mechanism of an existing install, or do you need to reinstall.

4) GUI vs. command line: Does this distribution give you an effective GUI method to do system config and maintenance operations that experts prefer to do via command line? Most people giving advise here think that GUI availability is a disadvantage. If you have the GUI you may not learn the command line. If you need to do such system tasks in some advanced way and/or very often, the command line is better and not learning it makes you a worse Linux user. I don't happen to believe that. If you choose to learn the command line way, having the GUI alternative doesn't get in your way. If you choose to use only the GUI, that may be best for you.

Almost all that source code would work across distributions if someone compiled it for your distribution. If they didn't and if you care enough then you could compile it yourself.

A distribution, like Ubuntu, that smoothly leverages a super upstream pool of compiled apps (Debian) gives you easy access to almost everything already compiled for your distribution. Use the package manager and almost everything is automatic.

Fedora (and in general the Redhat family) relies on a different pool of pre compiled stuff, and that might not be quite as simple to get access to pre compiled everything as with Ubuntu/Debian. But it is still fairly easy and a very massive pool.