Katra
It may sound like a tasty dish or the name of an internet-famous cat…but katra is The Search for Spock's symbol for the soul, mind, or essence.
Actually, it's not so much a symbol for the soul so much as it is the Vulcan word for soul.
As Sarek tell us:
SAREK: [Spock] entrusted you with his very essence, with everything that was not of the body. He asked you to bring him to us and to bring that which he gave you, his katra, his living spirit.
So katra represents the soul in the movie and apparently the afterlife works like a P2P network, because Spock is able to upload his soul into any brain he wants, as long as he has a hardline. What else is there to say?
iSouls
Well, we need to consider what type of soul we're dealing with because, as we know, not all souls are created equal. Is this soul really only a material consciousness, existing solely as synapses firing off in the brain? Is it the type of soul that is part of a great universal soul, returning to it after death like a raindrop in the ocean? A ghostly soul garbed in chains that doles out life lessons to miserly Scrooges?
No, nope, and not quite—although that last one would have been great.
Instead, The Search for Spock proposes a dualist notion of the soul. As Mahesh Anath explains it:
Star Trek embraces the idea that the mind can exist distinct from the body—at least for periods of time. Within the philosophy of mind, the mind's ability to exist independently of the body can be explained by the view of [mind-body] dualism. (Source)
From this philosophical perspective, the body is material, consisting of matter all the way down to our base molecules. The soul, on the other hand, is made up off…ectoplasm? It's made up of something that can't be reduced to carbon and oxygen elements.
But the two are capable of interacting. Philosophers have spent centuries considering how this interaction takes place, but in the Star Trek universe, we see this is possible through Vulcan superpowers like the mind-meld.
The Search for Spock takes it a step further by showing that the body can exist without the soul…but only as an empty shell. We see this in Spock's resurrected form. This body lives on Genesis and grows from an infant to an adult in a few days, yet it never develops a consciousness, lacking the essential "I" of personhood. McCoy confirms this when he first examines Spock's new body:
MCCOY: Rapid aging. All genetic functions highly accelerated.
KIRK: What about his mind?
MCCOY: His mind's a void. It seems, Admiral, that I've got all his marbles.
As such, The Search for Spock throws away reductive materialism—a ten-dollar way to say only the material world is real and our sense of self exists through activities in the brain. (Source)
Spock's had a brain this whole time, and it has been active. In fact during his pon farr, his brain was liking in overdrive. Yet until that body was united with his katra, no one was home.
Even with all this said, the film is very careful to not discuss any type of mind, essence or soul in relation to humans. When it comes to the afterlife in Star Trek, only Vulcans have a guaranteed ride.