Galactic War is unlocked once you have reached level 40.
Basically you can utilise your whole squad (5 at a time) to progress through as many battles as you can. So if your first five die, you can use other members of your squad to continue.
As you progress through each level the battles generally get harder, but the rewards get better as well.
As well as items and shards you can collect War Tokens, these are redeemable for character shards by clicking the 'redeem' button in the Galactic War section.
If you want to start a Galactic War over, then simply press the reset button in the Galactic War menu page. You can do this once per 24 hours.
You get to see the whole set of characters you are facing for each battle so you can plan your team accordingly. The battles take on the same format as the rest of the game, team vs team, so the hints and tips we have provided on our battle tips page would be useful here too.