War is an essential tool in the arsenal of your world.
In order to wage war, go to the world you want to conquer and click 'Declare War'. You may be in three offensive wars and three defensive wars at a time.
N.B. Wars are always declared in the defending world's home sector. Make sure you have a sufficient force in their sector before declaring war, or they will easily win.
Attacks are possible every 8 hours per war. When attacking, the fleet powers of you and your enemy are calculated. This takes into account the numbers of ships you have, your training and your weariness. Damage is obtained from a comparison of these two values, and ship losses are then calculated in proportion to how many of a ship type you have.
For example: if the damage is 10, and you have a fleet of 4 corvettes, you'll lose both those corvettes. If you have a fleet of 10 fighters and 2 corvettes (equivalent base power to the other fleet), you're likely to only lose 5 fighters and a corvette. Seeing as fighters are more quickly and cheaply replaced, it's in your interest you maintain lower tech ships as fodder during battles.
You win a war when your fleet power is more than ten times your enemy's, either before (in which case you'll automatically gain victory without fighting), or after a battle. When you win a war, you gain 1/6 enemy's GDP, 1/2 enemy's growth (if above 0), and 1/2 of budget and all resources.
You are protected from war in two cases: for one week upon joining the game, and for 5 days after losing a war. You may voluntarily exit war protection by either declaring war on a world (in which case you're obviously ready for war), or by sending ships (if you can transfer your fleet, you can fight with it).