What is this Galaxy At War stuff on the main menu?

Galaxy-at-War is Mass Effect 3's multiplayer. That's not the important bit - what you should be interested in instead is the "Galactic Readiness Rating". The idea is that the Reapers are invading, and the Galaxy is not, well, ready. How the main storyline resolves depends on the Galactic Readiness Rating when you go on the final mission.

There are three ways to increase Galactic Readiness Rating:

• Winning matches in Mass Effect 3 Multiplayer
• Using one of the Mass Effect 3 iOS apps (there are two: Infiltrator and Datapad)

Here is where you can find more details on how to increase readiness. Additionally, the Mass Effect Wikia has a little more on it.

That screen represents all aspects of Galactic Readiness, and will default to 50% for all Mass Effect 3 players.