Note: Since most of my experience comes from a long time ago when I played on easier difficulty settings, this might not appy on hard difficulty settings.
Some terms:
Empire: Large race with 8+ star systems (Largest Galaxy)
Kingdom: middle sized race with 4-7 star systems
City State: 1,2 or 3 star system race.

Note: I always play as Humans so I won't list them here as I have close to no experiance with their AI.
The Empires
Sakkra: Most formidable of all, tend to spam star systems and population.
Silicoids: can grow to be quite powerfull
Psilon: Not very powerfull early game but can dominate with tech late game.
Klackons: Can dominate with huge fleets of large ships.
Kingdoms
Alkari: Tend to be middle sized and decidedly average.
Trilarians: Tend to be middle sized and decidedly average.
Darlock: Tend to be middle sized and they also like to annoy you with spies.
Mrrshans: Don't prioritize reaserch which often make their ships quite weak. Sometimes they conquer something, but not too often.
Bulrathi: similiar to Mrrshans.
Gnolams: Tend to be middle sized and decidedly average.
City States:
Elerians: mostly camp in one star system. Then they threaten someone and get mauled as their ships are low tech.

Note: you can argue with these in the comments