Alliances are groups of Corporations (Guilds in many other MMOs) and all are player led groups. Some alliances are based off other online communities, eg Goonswarm (Something Awful) and Dreddit (Reddit). You train up a few skills and then you can create a corporation. You can then join an alliance if you want. It also costs ingame money to keep the alliance going.
I was an alliance director in a small alliance (10 corps, 500 members) and it's hard work. Dread to think what it's like running CO2.
Dreddit is the corp, TEST is the alliance. IIRC (it's been a loooong time since I played Eve) Dreddit was at one point the "main" corp in TEST but that might have changed.
I was an alliance director in a small alliance (10 corps, 500 members) and it's hard work. Dread to think what it's like running CO2.