The lore in the souls games has a lot of interpretive areas to it, so I can only sum up the hard facts to cover your bases if any of the big names get dropped in the third game:

Undead: Humans that are cursed by the Dark Soul (cover that later). They revive over and over again until they lose purpose and go insane, essentially becoming zombies. The player character in the previous two titles was an undead.

First Flame: The characters in this universe treat this object as though it is the glue that holds reality together. The Dark Soul (it's coming I promise) and three other immensely powerful souls came out of it, basically changing the course of a previously unchanging history as the inheritors of those souls shaped the world we know in the game. Now that the souls are out of the flame however, the only way to keep the flame going it seems is for people with immensely powerful souls to sacrifice themselves to it to act as fuel. This is an option offered to the player character in both titles, but even if the player doesn't choose it, the implication is that before the flame dies out, eventually another being comes along and does it.

Dark Soul (tadaaa): One of the four souls originally inside of the first flame. Whereas you can think of normal souls like some type of anime style mass of energy (you can use them to gain power, or shape them into objects of power - and they exist in all life) the Dark Soul is unique. The Dark Soul exists in fragments, inside all humans, and incubates inside them like a virus. Whereas the soul is life, the dark soul is often times considered the source of negative (and positive occasionally) human emotion. The dark soul drives non-humans bat-shit insane if it infects them, and can manifest as a location (called the abyss), sentient beings, and as little tiny fragments known as humanity. The general idea is that if you aren't a human with an incredibly powerful soul, too much dark souls wrecks your shit. Only immensely powerful and determined humans stave off hollowing and the corruption that seems to coincide with Dark Soul overdose.

I think those are the big 3 that are going to get tied in to the main story in more than just a referential way. Players are still looking for answers as to why the first flame being linked is so important and why we have the undead curse to begin with.