I've only done one EF, but this is what I gathered from my experience:
First off, a full alliance is helpful, but not needed. Four NMs spawn and they're very tough to kill unless you have either a lot of people or a good strategy. Next, the level cap for the fight (more on that later) is not the same as the minimum level, I think it's 5-10 levels higher (I know Buburimu's cap is 30, but I don't know what the level caps for the other areas are).
To start, you must have enough people in each party (4 if your home country is in 3rd, 5 if 2nd, 6 if 1st). You must also all be above the minimum level, and all be rank 3 or higher from the same country. You must also be talking to a gate captain (W.W., T.K., or I.M.) inside your home country - outposts or consulates don't count, and neither do Jeuno's Ducal guards. After everyone in your party gets permission to go on the EF, talk to the guard again to get teleported to the target region. Everyone will also get a gate glyph that can be used to return to town when you're done.
If not every party has enough people, have one party go, have some people use their gate glyphs, then have them disband and join the other parties so they have enough people to go (for example, if you need 6 people per party but only have 17 people, have the first two parties get permission and get teleported, then have one person warp back with the gate glyph, disband, and join the party that was left behind).
Next, search the whole zone until you find a "Beastman's Banner". It's a flag with the beastmen's signa on it and can be targeted. It does not show up on Wide Scan. After one person finds it, everyone should head to the flag and buff up. When everyone is ready, one person touches the flag to spawn the NMs. Four beastmen NMs will spawn, and the alliance will get a level cap effect. They will be the same race as one of the beastmen found in that region (so Kolshushu region will have yagudo and goblins, Qufim will have giants and goblins, Zulkheim will have goblins, orcs, and quadavs, etc). They will also have 4 random player jobs chosen from the different jobs those beastmen can be (so yagudo can be BRD, SAM, NIN, SMN, WHM, BLM, and MNK, etc). Once the NMs die, your country will gain rank points for that region. Everyone must then touch the flag again to get rid of the level cap status. Another flag will spawn elsewhere in the zone, so start hunting for it.
In order to win an EF, your home country must control that region at the next conquest tally. If your country does control that region, you get some CP - I heard it's always 3k (and that's how much I got for a successful Kolshushu EF), but I don't know how true that is.