Put basically, Ultimecia is a time paradox, desperately trying to alter fate. Technically, she does not exist. She is everything and nothing. The entire plot of Final Fantasy VIII is an unending circle, played out over time and space.

SeeD triumph over Ultimecia. Squall and Ultimecia go into the past, Edea receives Ultimecia’s powers. Squall gives Edea the idea of SeeD and Garden and travels back to HIS present reality. Edea is possessed by the Ultimecia of HER reality. Events of the game. SeeD triumph over Ultimecia. Rinse and repeat.

Why all this? What does Ultimecia wish to accomplish? Simply to survive - to do this, she has to destroy SeeD or more specifically the “fated children” (Squall etc…), the ones destined to kill her. Time Compression is the edge of the coin that lets Ultimecia manipulate the flow of destiny, in an attempt to change her fate. Anything is possible during this event.

“I am Ultimecia. Time shall compress. All existence denied!”

Each time she is defeated, Ultimecia goes into the past to give her powers to Edea, so that she can try again in that timeline, remember she dies in the past, not in her future hence she technically still exists there (If you built a time machine, went to the past and died while there. You would still exist in the present. The future is Ultimecia’s present). Depending on how you look at it, there is only one Ultimecia or an infinite number of her, transcending time – due to the nature of paradoxes both are theoretically correct.

Obviously SeeD have never failed to stop Time Compression, since if they did, reality as it is known would come to an end (past, present and future). Replaced with a world where only Ultimecia is able to exist.

As for who Ultimecia is, well… simply Ultimecia. Not Rinoa or Julia or anybody other than… Ultimecia. The main reason being what Fallen_Angel said.

You want a game that’s more complicated with this kind of time talk? Try Shadow of Memories/Shadow of Destiny. ^_^