What does the ending to Dark Souls 2 mean?

This answer covers only the original ending - not the SotFS edition

Well, this is all speculation, as is every theory in Dark Souls, but here goes:

NG uncovers that there are only 4 "Great Souls", and the 5th; the "Dark Soul". Each linking to the original owners. Old Iron King has the Old King's Soul, (Gwyn), The Lost Sinner has the Old Witch's Soul (Bed of Chaos), Freja has the Pale Drake Soul (Seath), and The Rotten has the Dead One's Soul (Nito). Meaning that these 4 beings are not the original owners of their own power, but found these souls and gained power through them. The 5th soul, the Dark Soul, was actually shattered (explained in the Dark Souls 1 Lore to be Humanity) which is why Nashandra is a) so easy to defeat and b) doesn't have an extra soul drop in NG like the rest of the main bosses.

If we follow the story of Dark Souls 1, which is very similar to that of Dark Souls 2, (flame dies, hollows come, someone links the flame, cycle starts over). No one single soul, however great, is powerful enough to complete the task of linking the flame. That's where you (the Chosen Undead) come in. You have the strength, and power to combine these Souls, in the attempt to rekindle the flame.

Now, why does Nashandra want the Throne? Why is Vendrick husked out and wandering aimlessly in a crypt? Why didn't he link the flame?? Well, subtle hints and miniscule pointers lead to this: Vendrick saw through what Nashandra was (Manus) and sought to thwart her attempts to seek the throne. He locked his soul away, accessible only to a Human, and then his loyal knight, Velstadt, carried him to the crypt, and kept watch over him, waiting for someone worthy to take the King's place. If you look closely, you can see the King's Ring sitting atop a pile of something (Rubble? Bodies?) it is actually the Kings armour. This was in hopes that whoever might take it, would leave the king in peace. (Had he not suffered enough?)

With the addition of the Three Crowns DLCs, you can now use the Giants Kinship on the armour you find, and actually talk to Vendrick himself, before he hollowed. He reveals that part of the reason he left the world to "decay" was because it was inevitable, and this course of action (he believes) is the only way to break the cycle of the flame - if it is rekindled, it is inevitably going to die again. But if it dies, it (should) stay dead.

Now, back to Nashandra. The King originally never knew of the Throne of Want, but Nashandra did. She was seeking power, and what could be a greater source of power than the First Flame? She manipulated the King into finding and destroying the giants, to find the keys necessary to uncover the Throne. This is why everyone is striving to have someone else get there first, to sit upon the throne before Manus (Nashandra) can do so.

So it's pretty easy to tell that the Throne of Want is in some way linked to the Kiln of the First Flame; ash on the ground, the Throne is inside a kiln-shaped housing. It's where you can either choose to link the flame, or leave it and usher in the dark. Being of both Fire and Dark, it should be up to you, right? So why do you not get the choice to leave? Some say it's for the common good, some say it's because you are a pawn that really has no clue what's going on (I say jump - you say how high?), That's still very open to interpretation. But I think the choice is still there, just not openly accessible by the player. The Chosen Undead sits on the throne, the kiln closes..... but no flame...? I think one of two things are happening here: a) the choice happens after the credits roll, adding to the mystery of it all, which sort of ties in with the choices in DS1 (but I didn't link the flame? Why are they saying someone did?) or b) to break the cycle; the power to revive the flame is locked away forever, and thus the cycle is broken.

However, on some further speculation, a common beleif is that the final cutscene is the choice coming full circle, as it is destined to, again and again. Even if you didn't link the flame in the original, it would only be so long until someone else did. Possibly a nod from the developers saying "You know what happens next, so we're going to end it here before it gets too repetitive".

I hope that sheds some light on the situation.