From: Vermont, USA
I should note that we are planning to release an updated demo by next week. In the meantime though, the way the turn-based combat works is as follows:
Each combat round you have a certain number of Action Points (APs). Each action costs APs. For example, you may have 6 APs and it could cost 3 APs to lock on and fire a missile. It may cost 2 APs to fire your laser, perhaps 2 or 3 APs to actively use ECM to try to jam an enemy missile, or 2 AP to focus your shields, etc. You have to choose each round how to use your APs given the mix of actions. Some actions give you lasting benefits, for example for 1 AP you can taunt the enemy which can affect his accuracy, that can last for several rounds. When you've used up your APs, you end the round, the enemy now gets his round and then it goes back to you.
The various improvements you can make to your ship and to your own abilities can give you more things to do with your APs, various special abilities, or can increase or decrease the AP cost of some of the existing actions. For example, you can upgrade to a turbo-laser that can fire with only 1 AP cost, but has a chance of overheating, you can get a much more effective ECM system but it may take more APs to use, and so on. Each ship type has different upgrades, each career has different options for skill/ability upgrades as well, so battles can end up quite different depending on the choices you make both before and during the battle.
As you get to bigger ships too, things like crew composition and boarding actions come into play as well.