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EVE Online is what you'd get if you combine Traveller with Microsoft Excel, Euro Truck Simulator, and a cliff notes version of Machiavelli's The Prince. Developed by Icelandic studio CCP Games, the company that briefly bought White Wolf with delusions of making a World of Darkness MMO before realizing that building an actual mmo is much harder than making spreadsheets fly in space.
EVE is set in the distant future after humans left Earth through a giant wormhole which subsequently closed, stranding them and causing a dark age for the newly formed colonies until they were able to develop back to the level they'd been at, whereupon they immediately set to fighting each other. Technology wise it's somewhere around the level of Star*Drive or Babylon 5 where big ships are able to jump around unassisted but smaller ships need some help to cover interstellar distances.
There are four main NPC factions in the game, who's role is generally to represent different combinations of technology.
Theocratic empire, loves heavily armored ships covered in gold filigree with lasers. Amarr ships are regarded as somewhat one-dimensional with very tough armor resists but limited damage choices. Their capital ships are the preferred meta because armor tanking offers a much deeper buffer than shields and in capship slugfests it's more important to survive long enough to receive healing.
From the tiny Punisher frigate to the giant flying assplug that is the Avatar titan, Amarr ships basically fly the same. They have lasers, they have tough armor, and they're gold. However, if there's one ship of note it's their freighters, which look like the rebel transports from Empire Strikes Back.
Capitalist plutarchy, their ships are unpainted metal boxes with missiles and railguns. Caldari are shield tankers, which makes them a very solid choice for subcapital combat (armor plates have a significant impact on ship mass, shield upgrades don't), but highly situational for capital ships. Their one "good" capship is the Phoenix, the lone dread that can actually hit anything while moving on account of it being a giant metal box of missiles.
The archetypal Caldari ship is the Drake battlecruiser: a lumbering brick encrusted with missile launchers and capable of being shield-modded to rival a battleship. A new pilot can amass the skills to competently use one in less than a year. The combination of missiles, low-ish skill requirements, and tough defenses makes the Drake a go-to design for fleets using low-skill player swarm tactics.
Noblebright democracy, flies curvy ships with drones and blasters. Gallente kinda suck; blasters are short ranged and power hungry, while drones work slower and can be destroyed. They're armor tanked, but with an emphasis on active self-repair (which is shit) as opposed to having a really beefy buffer (which increases the chance you'll survive a single alpha strike and be healed). Generally regarded as a PvE choice on account of having the best drone/fighter carriers.
Gallente ships have a bipolar tendency, which can be illustrated by looking at two of their cruisers, the Vexor and the Thorax. The Vexor is a slow, tough, amorphous blob that can carry a lot of drones, but will mostly sit around watching things very gradually die. The Thorax resembles a flying penis that wants nothing more than to get to point blank range and violate the enemy with neutron blasters. Most Gallente ships fall into one of these two camps.
Tribal republic of dirt poor ex-slaves, fleet made out of rust and duct tape and armed with guns. Minmatar ships benefit from not needing to use capacitor power to shoot, allowing them to focus on engine boosting. They also can apply all damage types by switching ammo. Being fast and flexible, Minmatar ships are often the meta for their role, at least in the subcapital categories where costs are low enough that survivability isn't a serious consideration. Their capital ships however are garbage and nobody pretends otherwise even ironically; they just don't have the toughness to slog it out.
While the Minmatar have many ships of note, none are more iconic than the tiny, unassuming Rifter. More rifters have been flown and lost in EVE than any other class of ship. Small, fast, and cheap, they are built in the thousands and thrown onto the field to pin down other bigger ships and keep them from escaping. Fly a rifter, die gloriously, and then grab another rifter.
The real action is between player organizations. The basic building block of organization in EVE is the Corporation. The corp is equivalent to the guild or clan of most MMO's, with a member list, some super users, guild bank and assets, etc. However, corps can have more assets, such as deplorable space equipment that remains on the server indefinitely (as long as the bills are paid, resources are supplied, and nobody blows it away). The typical corporation has a few dozen to a few hundred members, although holding corps tend to be very small and actively recruiting player corps can range into the thousands of members.
Corporations can band together formally into alliances. Corporations in an alliance can make sovereignty claims over systems by deploying and maintaining a thing in space that says they have sovereignty. Having sovereignty over a system allows deployment of system upgrades that can make the system more ideal for the owner's intended use. The typical alliance has thousands of members, with the largest breaking into tens of thousands.
Alliances often informally band together into coalitions, usually consisting of a central power with satellite allies. Coalitions aren't precisely modeled in the game, they're more of a meta feature, although alliances can set standings towards each other so their pilots see other pilots as friendly or hostile. The major hallmarks of a coalition are having a unified command and communications so that all coalition members can cooperate both at the battle and campaign level of operations.
An actual list of player coalitions in the game would be impossible to maintain as they change frequently and often without warning. However, as a general rule of thumb there are always multiple predominately American coalitions that hate each other, and one Russian coalition that hates everyone who isn't Russian.