Combat in GalCiv is pretty simple. Each ship has three important factors. Attack, Defense, and Hit Points. Each ship takes turns shooting at the other ship, with the attacker always going first. Each shot is the done as two random numbers, the shooter's number is between 0 and his attack value inclusive, the target's number between 0 and his defense value inclusive. If the attacker got a higher number, the defender looses a number of hitpoints equal to the difference between the numbers. If the target survives (has a positive number of hit points remaining), the target becomes the shooter, and the cycle continues. There is one common exception to this. If the defender (not the target, the ship that was sitting still and got jumped) has a higher defense rating than attack rating, half of the defense rating, rounded down, is added to the attack rating for purposes of the combat.
Now that you should have a handle on what the numbers are used for, you can decide what improvements you want to pursue to increase these values. Mining a military resource will improve the attack and defense value of all ships. (I seem to remember that it also improves the base defense value of star bases, but not the value added by defensive modules). Certain technologies will improve your attack and/or defense ratings. see http://webpages.charter.net/trevorpowdrell/a_weapons.htm for a list of all techs that will improve your weapons ratings, and http://webpages.charter.net/trevorpowdrell/a_defense.htm for a list of all techs that will improve your defense rating. Furthermore, as a ship wins combats, it earns experience, which will raise the level of the ship. I'm not sure what the level affects (I'm playing the first game where ships are fighting enough and surviving to level up right now), but I'm pretty sure it improves the defense of a ship, and possibly the attack. Ships in orbit of planets also get their attack boosted, and the boost is greater depending on the planetary defense social projects on the planet in question: http://webpages.charter.net/trevorpowdrell/a_planetdefense.htm
Then there's planetary improvement that improve the quality of starships. http://webpages.charter.net/trevorpowdrell/a_starshipquality.htm has a list of all the social projects (buildings) that can increase the quality of starships built at that planet (but not at other planets, keep that in mind). Starship quality is reflected in increased hitpoints. There are also technologies that directly improve the hitpoints of ships: http://webpages.charter.net/trevorpowdrell/a_hitpoints.htm
Now, another thing that can affect the attack/defense rating of a ship is being in the same sector as one or more starbases with military modules. There are modules that increase the attack of all ships in that sector at http://webpages.charter.net/trevorpowdrell/a_attackassist.htm and the one module that helps defense at http://webpages.charter.net/trevorpowdrell/a_defenseassist.htm
Finally, there's one more value that hasn't been mentioned yet, and that is repair rate. Basically, each ship can recover a small percentage of its total hitpoints each turn. I don't know what the value is at first, it could be as low as zero, but there are many ways to improve it. First, certain techs (and one trade good) improve your repair rate: http://webpages.charter.net/trevorpowdrell/a_repair.htm
Second, there is a starbase module that also improves the repare rate of all ships in the same sector: http://webpages.charter.net/trevorpowdrell/a_repairassist.htm
Finally, it is possible, though disputed, that being in orbit of a planet improves your repair rate.
I can't promise that all of this information is 100% accurate, and I certainly won't promise that it's complete, though if anyone wants to make corrections/additions, I'm all ears.
If you go to the metaverse page, you'll find a tab on the left labeled "simulator." That's a combat simulator that will let you get a feel for what chances an attacker has against a defender, all other factors considered equal.
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