You Know Who asks,
In Back To The Future (the first movie), Marty McFly (for sake of this question, Marty I) goes back in time to 1955, spends a week there, alters the past, and returns to 1985 just in time to save his friend, the inventor Doc Brown. When he returns to 1985, Marty I gets there just in time to see himself (let's say Marty II) driving the time machine around go back in time to 1955 after escaping from the Libyans. Now Marty II grew up in the altered universe created by the changes that Marty I made in 1955 -- when Marty I goes home, his family is well off, his father is a successful author, he's got a big new truck, etc., all the fine trappings that Marty II grew up with.Very interesting question.
So what happened to Marty II? He went back to 1955, but when did he return? Is he still in 1955? Did Marty I replace Marty II? Was that fair, moral, and just? Marty II, having had a different childhood than Marty I, would not necessarily know to do the things that Marty I (perhaps) already did in 1955 to alter history and get his parents to meet since he was never aware they had any problems to begin with. Does he go and watch Marty I do them himself? Does he encounter Marty I when he returns to the same place and time as he did (presuming the flux capacitor is set to the same date/time?)
In the Second Movie, Marty I goes back to 1955 again and watches himself at some distance do the things he did in the first movie - where's Marty II? The guy he's watching doing these things is clearly Marty I from the first movie, and not well-adjusted Marty II. Marty II has not somehow rejoined his own life, since it is clear that it is Marty I who is now enjoying Marty II's sportin' lifestyle at the start of the Second Movie.
First, let's talk a little bit about time. Think about the sort of trail that a moving object leaves in a time-lapse photograph tracing out its trajectory. We could extend that notion to an object in real life leaving a trail behind it as it moves (I'll assume this concept is not foreign to you). If we view all of space and time as one big spacetime, then a person's path through spacetime is a person-shaped tube. We call that the person's worldline. When you bump into someone, your worldlines intersected.
The length along your worldline is the time you experience. Say you are playing hockey and someone bumps into you sending you into the boards. The distance measured along your worldline between the points where your worldline intersected your opponent and the point where your worldline intersected the wall is the time it took between the two events. That time is absolute and we call it the observer's "proper time".
Now we can talk about the time between two events. It turns out that we can only say in an absolute sense that one is later than the other if it is possible to send a light signal from one to the other; if not, then different observers will place them in different time orders. Those that have an absolute time order are called causally related because what happens at one can cause something at the other.
The universe is the set of all events, that is, places at a time. Our worldine is comprised of al the places at a time that we have been. What is interesting about time travel scenarios, like we see in Back to the Future, is two things:
First, that you have a person whose worldline touches two events E1 and E2 which have an absolute time order such that E2 is objectively later than E1, but which are experienced by the person in the opposite order. For example, Marty sees his parents meet after he was born, but of course, the birth had to have come first. This means that Marty's worldline not only curves in a way it shouldn't, but also that it is experienced as smooth, but has a discontinuity in it. His worldline stops and then starts up somewhere else, but his sense of proper time is undisturbed.
Second, those two events are causally related, meaning that what happens at one can effect what happens at the other. As a result, and this is where all the paradoxes are generated, what the person does at E1 could cause events at E2 to be radically different or not to have happened at all.
Marty's brain has the state it does, he remembers what he does because of his interactions with the universe. Each of us is in some sense a partial record of what has happened to us. What is peculiar in the film is that when things conspire to create a quite different future in the single existing universe, he loses material aspects (photos begin to fade, muscle control fails), but his memory, which is also a physical manifestation of his personal past which is the supposedly soon to be different future does not.
Since the universe is the set of all events, by changing events Marty could be seen as having created a different universe. Marty I at the time he wakes up in his clothes is in one universe whereas Marty II when he wakes up in his clothes on a parallel day at a parallel time is occupying a different universe, yet he is in the new universe with a brain that bears the marks of experience from either a universe which doesn't exist (which is problematic as something that does not exist cannot have physical ramifications) or a different universe. If we demand consistency and say it is a different universe, then we have a sense of time, Marty's proper time, that exists outside of spacetime, but time is supposed to be something which is an aspect of spacetime. Therein lays one aspect of the weirdness.