In the "Eastern" section of the galaxy lie a couple of empires that, while not the strongest in the galaxy, are in fact the fourth and fifth strongest in the galaxy as far as non-fallen empires are concerned. One of these (the fourth most powerful) the United Nations of Earth --sprawling stars of the Human species, dedicated to individualism, their democratic utopia, and being friendly to alien species (if reluctant to ever actually allow Migration treaties). Their fleets are not the galaxy's most powerful, but they are certainly one of the heavy hitters... but it's not the ships that the United Nations' primary strength lies in. Where the United Nations excels, despite occasional periods of intense military research, is industry. The United Nations had for the first century of its existence focused almost entirely on industrial technology, to the point that in 2275 they were still fielding outdated fusion missiles. But the positive consequence of this is incredible industrial output, used to fuel growth and constant, utterly unlimited building. The United Nations are typically able to trade for whatever they want, thanks to handing out thousands of minerals or energy every time they want something without even a second thought, and in war resources continue not to be a problem. The United Nations have never actually conquered a planet, preferring instead to forcibly convert their enemies to democracy and at least some level of xenophilia. This is seen in their vassals -- the Allied Ytrellan Worlds, once the tiny, genocidal, and wildly xenophobic Ytrellan Empire until the United Nations converged on them, up until its recent integration the Democratic Hythean Union (a xenophobic newborn empire that sprung up in Human space), and the recently-formed Confederation of Nodox, a planet from the rival Kithri Unified Planets that was "liberated" after an encroaching frontier station was placed by the KUP. Much more peacefully a protectorate are the Rethellian Union of Suns, bordering the Confederation of Nodox (and a factor in the Confederation's conversion to freedom), which sprung up into infancy and simply asked the United Nations to protect them.
The other great empire of that sector of the galaxy is the Nation of Haspalv'Mina (the fifth most powerful in the galaxy), home to the warlike but xenophilic empire of the Haspalv. It is not as large as the United Nations, but has a similar number of worlds crammed into that space, and has a fleet that usually somewhat outstrips the Human fleet. Where the Humans focused on industry, the Haspalv focused on reaching the cutting-edge of war and never stopped. They have conquered most of the Rutharian species (former Kingdom of Ruthor), but are now oft led by them, thanks to the Haspalv focus on democracy and love for other species. The former Kingdom of Ruthor, rival to both them and the United Nations, was half conquered by the Haspalv and half vassalized, with United Nations help.
Other much more civilizations in that sector of space are the Wessari Confederation, a civilization of extremely peaceful traders which has lived with lukewarm but not ultimately hostile relations with their more warlike neighbors the Humans and Haspalv, the Allied Rutharian Suns, a product of United Nations desire to convert their rivals that ultimately did not end up in vassalage, and the Pithok Republic, a baby free nation that is the protectorate of the Wessari but in Haspalv lands, and the Bimathi Guardians, a Fallen Empire built around religion -- they do not get involved in the world's politics, but they provide a compelling reason to leave Tomb Worlds and their rare Holy Worlds alone.
The Haspalv and the Humans form the backbone of the four-nation federation the Alliance of the Free Swords, which has formed in part due to the longstanding friendship of the Haspalv and the Earthlings, and in part due to opposition against the less free empires of the galaxy which have oft served as either threats (for the weaker members) or obstacles for the power of the stronger members (the Nation of Haspalv'Mina and the United Nations of Earth). The individual members may not be THE strongest in the galaxy, but their federation is certainly by far the most powerful federation there is. The Alliance of Free Swords is formed of the three ideologies of individualism, love for other species, and lack of fear of war, and this tends to inform their politics.
In the more "Northeastern" section of the galaxy lie the other half of the Alliance of the Free Swords. These two nations are the Democratic Adeex Planets (highly free, to Earthling approval, and yet militaristic to Haspalv approval), an empire with deceptively many worlds but ultimately not quite the raw power of the Human or Haspalv, and the Figyar State, a highly militaristic and somewhat materialistic empire which is the weakest member of the federation. The Figyar and to a small extent the Adeex are members of the federation originally because of a common rival of all four of the nations of the Alliance, and what lies squarely between the Adeex, Figyar, and the Humans -- the Kithri Unified Planets.
The Kithri Unified Planets are a mildly xenophobic but highly ambitious Star Empire which once provided a threat to the Earthlings but thanks to wild growth of the Humans and Haspalv and occasional wars, the Kithri are now one of the weaker rivals of the Alliance. They have two vassals (the newborn Canthari Unified Planets, who actually sprung from the edge of Bimathi space), and the Grand Duchy of Birgas, neither of which is actually strong enough to really tilt the balance of power. The reason the division of the Kithri has only just begun with the Nodox Confederation and will ultimately proceed slowly is because the Kithri have a very powerful ally indeed -- the Sondrithan Accord.
The Sondrithan Accord is a free nation, but a xenophobic one -- and it has been placed opposite the Free Swords thanks to its choice in alliances and now genuine rivalry with the Humans. They are also the most powerful empire in the galaxy, a truly sprawling mass that makes the United Nations look somewhat small, and with at least four and perhaps five vassals by force, and with military that dwarfs that of the individual members of the Free Swords. Their alliance with the Kithri keeps the Kithri mostly safe, and it's only their frequent conflict with their neighbors the Photecian Commonwealth and their own occasionally rebellious, oppressed vassals that keeps them from bringing their full force to bear -- a battle that would be perilous indeed.
Only a couple more nations are really notable from a human perspective -- the second-most powerful and formerly most powerful nation the Unified Menjeti Worlds in the "Western" segment of the galaxy, known only for their wildly tumultuous relations with their smaller neighbors the Tezekian Confederation, the third most powerful Photecian Commonwealth, which is friendly with the humans in part because of ideology and in part because of the presence of the Sondrithans... but which remains too far away to really matter beyond their wars with the Sondrithans, the Kel-Azaan Confederation, friendly neighbors of the Earthlings by way of their Ytrellan vassals, who despite frequent trade and beautifully warm relations run in different and rather neutral political circles, their rapidly disappearing collectivist rival the Associated Sek-Lokkar Systems, the other fallen empires such as the patronizing Trakpocian Ancients and the wisdom-obsessed Turnbator Chroniclers -- relations with them are actually more existent than with the Bimathi, but only in that they have made requests of the Earthlings to turn over their scientists or civilians and then been refused as the United Nations doesn't force its citizens to go anywhere.