From: New Zealand
Combat is at corps level, and depends upon various factors - troop strength, readiness and quality obviously, entrenchments & forts, terrain, gas, tanks, shock tactics spring to mind.
Artillery units bombard and inflict strength point and readiness hits, and can destroy entrenchments, and forts too if it is siege artillery.
Air power is important for recce & artillery - you get to see into a certain number of enemy held hexes, and artillery bombardment is increased in effectiveness into those hexes - plus there's the bonus of seeing what it is you are attacking or might be attacking you.
Combat usually results in one side vacating the hex, but occasionally it will be ongoing - units of both sides remain in the hex and will fight again next turn - but you can withdraw the ones in there and send new ones in before the battle is redone. This commonly happens in cities and forts, but occasionally in "open" terrain too.
One of the main things that, IMO, give GOA great "feel" is the HQ activation system - you have to spend HQ points to activate infantry units to attack the enemy - spending 1 point will activate every unit adjacent to the HQ you spent it at. That might sound like a lot - but HQ points are horrendously expensive - roughly the cost of an infantry corps' worth of equipment. You get a bunch at the start of the game and they have to be nursed because you will rarely get to buy more than 1 per strategic phase thereafter - if any!
I'm sure someone will add to this if I've missed any important points.