Wait? So we can actually control our ships? Must...resist...temptation... Gah, I am already having to fight off the urge to look at that unprotected save I Grabbed a while back from somebody else's turn...
Combat takes place in a separate game mode. Every ship and spaceborne unit in a single sector will be individually represented in a real-time battle. You have some control over your ships' artificial intelligence from the 'Empire options' menu by creating and/or modifying strategies and then strategies can be assigned to ship types or fleet task forces.
This is the first thing I found searching Y.T., it has a pair of carriers loaded with fighters beating up some ships and a station defending a planet. There is probably a bit of a technology discrepancy favouring the fighters, I would normally expect them to get hit a touch more often than that...
http://www.youtube.com/watch?v=I7ZJOXyKh9cNote that there is an option to view a sector, this is useful because it lets you place your units, provided that there are no enemies in the sector, you can use this to place your stations in a better formation, or to spread out a group of satellites so as to cover a larger area or something...