Blam's advice is spot on -- you need lots of Fighters against enemy Starships, and you'll want lots of Cruisers to take out their Bombers. You were playing against the Mad Bomber AI Type, which has the Bomber Starships as a unique superweapon. Be warned that as they get stronger, they get Mark II and then Mark III Bomber Starships, also. They also send larger waves than many other AI types -- however, they hardly defend themselves at all, so as long as you aren't dying yourself (which is always an uphill battle with that AI type), you'll have a comparably easy time with their planets. This is basically one of the "rusher" variants of AI.