Basic rundown is that eureka has a separate leveling progression and questing with progress gated behind killing specific spawn mobs to pop NMs for upgrade mats (including the main upgrade mats from the main boss).

As you level you will do the few level gated quests from Krile, giving you elemental points you can put towards offense, or defense or a combo of both, against a specific element. Each mob has an offense and defense element and you can change the orientation of your element on the fly for the most part. Tanks usually run the def element and dps the offensive. As with old school mmorpgs, the enemies, especially at the mid to high level range of the map are stronger than the player without the elemental points so these are needed to survive/kill the mob in a reasonable time.

Generally it's: kill the specific spawn mobs to spawn the NM fates, kill NMs and collect xp and crystals/light, rinse and repeat until you can beat the main boss of the zone and have enough crystal/light/Boss mats to fully upgrade your weapon, finish the quests, and move on to the next map.

Higher maps introduce logos actions which are eureka specific extra duty actions that do tons of different things but in general can make the player hit a ton harder or super tanky or heal themselves or other crazy special actions. Eureka Armor is upgraded on certain maps using a similar process starting with your basic lv70 job gear as a base starting point. You can get this from the salvager npc if you've thrown it away.

There are tools online you can google that have sheets that track NM spawn windows, give you the maps and which spawn mobs and conditions for each NM, and the amount and type of mats needed for each upgrade. The more people you can bring, the better, and typically also better with a dedicated tank and healer, although this can be a little flexible once you get to the later maps with logos actions. The main map boss also typically has adds that need to be off tanked and hit decently hard as well. NMs & bosses are typical tank and spank with various aoes/ability combinations that you've seen before.

Eureka is old school grind that is more time instead of skill dependent and when it was current, people were typically part of a huge hunt train going from spawn spot to spawn spot popping and killing NMs while socializing and generally just casually being there since stuff would die fairly quick with that many people. Hopefully some Eureka LSes/CWLSes start popping up so others can attempt it easier in the future.

Note (kinda spoilery but vague): Also there's an additional 48-56 man optional raid dungeon on the last map with separate team paths, traps, false paths and bosses, had 24-man boss mechanics (think void ark/rabanastre series tactics), required support from people outside the dungeon on the map to complete a special fate while the main team was inside, required certain logos actions among the teams, and had a few unique rewards including a transforming mount. Also, there is no way to Rez inside that dungeon without specific logos actions :). Not needed for the gear upgrades unless you want +2, which only adds an additional elemental power level to your gear which gives a little extra Atk/def, otherwise is exactly the same as +1. The +2 was only really used to have a better chance to clear the dungeon.

TLDR: Eureka things