Fridge Brilliance - Earth has very good diplomacy skill against the AI, but that doesn't help them against a human, right? Well depending on the player, when playing as a different alien race, they might see the Terran Alliance and decide "eh, they're cool" and put less espionage, etc into their faction.
Strictly speaking it doesn't help them much against a human — the greater your diplomatic bonus over an AI race, the more willing they are to bend to your whims. Since this is dependent on the difference between your and the AI's diplomatic bonus, Earth tends to be less susceptible to it.
Game Breaker - Psionic Beams. Some of the Neutral alignment bonuses are pretty overpowered too, especially the instant terraformer and research centers.
The racial bonus "Creativity" is possibly the worst offender, at the start of every turn there is a 5%-10% chance of automatically completing the current tech research regardless of time left, with the exception of the technology victory tech which it does not apply to, its prerequisites have this chance but down somewhere at 1%-3%. Considering that even on fast research settings with a tech focused race major techs can take 40+ turns to research this is boon. Additionally if the current research would be completed on the turn anyway creativity gives you the next tech free, including the final step for a Technology Victory... Praise unto the Random Number God!
The Thalan Empire was generally regarded to be the most overpowered of the playable races in the sequel's first expansion Dark Avatar because factories only cost a fourth to build for them relative to other races. The Arcean Empire's ability to always strike first makes them this to anyone who relies on glass cannons, as they'll blow their ships' weak defenses to hell before the glass cannon can return fire and also makes glass cannons very, very powerful for the Arceans since if they have enough firepower to kill the enemy ships in one attack they won't need any defenses.
The AI in the first two games tends to be very lenient about cultural warfare, so hijacking their worlds with influence starbases is a remarkably useful strategy and can save you a great deal of time. By the third game, it's wised up a bit and is more resistant to attempts to culture-flip its worlds.
That One Boss - Play a map where the Dread Lords have a substantial presence and watch as your early game colony ships, defensive (or offensive if you play that way) fleets, constructors, scouts, and freighters get brutally destroyed by flying chunks of daemonically possessed space coral. Actually wiping them out will be a long, painful, and economically ruinous process considering that their fleets will be entirely composed of what are essentially evil precursor ships. On the plus side, pretty much every other major and minor race you share your galaxy with will hate them and also fight them.
Dread Lord ships are also glass cannons; they do more damage in one shot than any player ship can survive, but go down a lot easier than you would think. Using glass cannons yourself lets you basically Zerg Rush them.
They also have a massive crippling weakness - if you shut down all their factories with spies, their fleet production will be slowed to a snail's pace. This is because as a minor race, they cannot spy themselves, meaning they have no way to rid themselves of a spy infestation. Of course you need to find the one planet they are on in the vastness of space, and you have no indicator of their territory using the starmap, so actually pulling this off against them requires foreknowledge of their location or pure dumb luck.